Topics — What a work is about. Nucleation holds whatever reading the owner files here, whether or not an Anthology of the SOTA topic could also hold it; instruction for ML practice belongs to the anthology (ADR-048). The words are seeded from the clusters the reading-time triage of 2026-09-25 found, and grow by decision.
Quantum foundations — what quantum theory says about the world — contextuality, epistemic restrictions, interference, reconstructions and toy theories, quantum thermodynamics.
106 of 2004 LIT documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| LIT-001 v2 | Quantum Mechanics can be understood through stochastic optimization on spacetimes | Lindgren & Liukkonen (2019), DOI-10.1038/s41598-019-56357-3. The paper derives a Stueckelberg-type equation i∂φ/∂τ = (1/2m)□φ − Vφ (Eq. 36) from an HJB equation, but only after putting the imaginary unit in twice by hand. The first i is a factor √g = i multiplying the Lagrangian; standard relativity uses √(−g) = 1. The second is a complex “variance” σ² = i/m (Eq. 29), chosen so that the Hopf–Cole substitution J = log φ linearizes the equation. The linearization itself is the standard λ = σ²·(control cost) condition. The Schrödinger equation (Eq. 43) then comes from c → ∞ plus complex conjugation. The Dirac equation is not derived, and the “Klein–Gordon equation” (Eq. 40) has 2mV where m² should be. | Rejected — read in full 2026-09-25 (NOTE-003); the claim that the i comes from the Minkowski metric depends on the non-standard √g, not √(−g). The complex noise it also needs leaves no real stochastic process or optimization behind. Nelson (1966), Guerra–Morato (1983) and, for the control-theory side, Kappen (2005) cover the real content better. |
| LIT-003 v2 | A structure theorem for generalized-noncontextual ontological models | Schmid et al. (2020), ARXIV-2005.07161. Take a tomographically local GPT and any convex-linear, empirically adequate, diagram-preserving map M from it into FVect_R. Then M(T) = χ_B ∘ T ∘ χ_A⁻¹ for unique invertible linear maps χ_A (Thm 4.1). Consequences: - Every diagram-preserving quasiprobabilistic model is an exact (non-overcomplete) frame representation (Prop. 4.3). - Every diagram-preserving noncontextual ontological model has |Λ_A| = dim(A), the GPT dimension (Cor. 4.2, §4.3): exactly d² ontic states for a qudit and 4 for a qubit. - The paper also proves (Cor. 3.5) that three things are equivalent in arbitrary compositional scenarios, without needing tomographic locality: a noncontextual ontological model of the operational theory; an ontological model (simplex embedding) of its GPT; and a positive quasiprobabilistic model of its GPT. | Active — read in full 2026-09-25 (NOTE-017); It is the formal backbone of the “classicality = generalized noncontextuality = positive quasiprobability” position that LIT-007 and LIT-019 appeal to. The main theorem is fully proved and short. Read §3–4 and App. B. §5.2 illustrates why tomographic locality is necessary, and its numbers should be checked (see corrections). |
| LIT-007 v2 | Quasi-quantization: classical statistical theories with an epistemic restriction | Spekkens (2014), ARXIV-1409.5041. Take a classical statistical theory over a phase space Ω = R^{2n} or (Z_d)^{2n}, and allow only uniform distributions over V^⊥ + v, where V is an isotropic subspace (a Poisson-commuting set of linear “quadrature” variables) and v ∈ V gives their values. The result is operationally identical to the quadrature quantum subtheory (the stabilizer formalism for odd prime d) for continuous variables and for odd prime d; the proof is that the Wigner functions of that subtheory are exactly these distributions (§IV.A). For d = 2 the two theories are isomorphic element by element but make different predictions, because the qubit stabilizer subtheory is contextual (§IV.B). | Active — read in full 2026-09-25 (NOTE-013); the clearest single statement of the epistricted-theory programme, and the source of the weak/strong nonclassicality split that 2111.13727 and 2005.07161 build on; read it for §II–IV and App. A, not for Table II, which it asserts rather than derives. |
| LIT-010 v2 | Quantum Thermodynamics | Vinjanampathy & Anders (2015), ARXIV-1508.06099. A 2015–16 review with no new results, whose one synthesis (§5.2, §7) is that the field used four inequivalent notions of work: ensemble ⟨W⟩ = ∫tr[ρḢ]dt (Eq. 2), two-point-measurement W = E_m^(τ) − E_n^(0) (Eq. 24), ε-deterministic single-shot A^ε ≈ −kT ln(Z_Λ*/Z) (Eq. 49) and resource-theory w^max_ε = F^min_ε(ρ_S) − F(τ_S) (Eq. 50), of which only the last two coincide. It also judges that the field had “only made a small number of experimentally checkable predictions of new thermodynamic effects yet” (§7, p. 39). | Active — read in full 2026-09-25 (NOTE-011); still worth reading as a short, accessible tour of where quantum thermodynamics stood in 2015–16, especially §3 (fluctuation theorems derived in full) and §5 (single-shot and resource-theory second laws). It is a map of the field, not an authority. It is dated: nothing after early 2016. Some formulas are garbled as printed (Eq. 68, the operator-sum statement, the Diesel strokes), so cite the primary papers it points to rather than the review. |
| LIT-016 v2 | The sheaf-theoretic structure of non-locality and contextuality | Abramsky & Brandenburger (2011), ARXIV-1102.0264. For finite measurement scenarios, an empirical model has a factorizable (local / non-contextual) hidden-variable realization iff its compatible family of distributions extends to a global section (Thm 8.1). Over the reals every no-signalling model has such a section (Thm 5.9: “negative-probability LHV” ⇔ no-signalling, because both span the same subspace, of dimension D = Σ_{U∈𝒰}(l−1)^{|U|}, Thm 5.4). Contextuality comes in three strictly nested strengths (probabilistic ⊋ possibilistic ⊋ strong), with the Bell, Hardy and GHZ(n≥3) models as witnesses of each level. | Active — read in full 2026-09-25 (NOTE-016); the founding paper of the sheaf-theoretic approach to contextuality. It is short, fully proved and still the standard reference for its definitions (empirical model, strong contextuality, the global-section criterion). Later work (cohomology, the contextual fraction, the CbD comparison) extends it but does not replace it. |
| LIT-018 v3 | Distinguishability and Accessible Information in Quantum Theory | Fuchs (1995), arXiv:quant-ph/9601020. Minimizing the Bhattacharyya overlap Σ_b √(tr ρ0E_b)√(tr ρ1E_b) over all POVMs gives exactly F(ρ0,ρ1) = tr√(ρ1^{1/2}ρ0ρ1^{1/2}) (Eq. 3.28). The minimum is attained by a projective measurement in the eigenbasis of M = ρ1^{-1/2}√(ρ1^{1/2}ρ0ρ1^{1/2})ρ1^{-1/2} (Eq. 3.67). For binary ensembles, the same Schwarz-inequality technique gives three results: a short proof of the Holevo bound I(t) ≤ S(t), with equality iff [ρ0,ρ1] = 0 (Eq. 3.149); a tighter upper bound L(t) (Eq. 3.187, integrated in Eqs. 3.230/3.244; closed form for qubits in Eq. 3.269); and a measurement-attained lower bound M(t) = tr[(1−t)ρ0 ln L_ρ(ρ0) + tρ1 ln L_ρ(ρ1)] (Eq. 3.203). The thesis also proves that two states can be broadcast iff they commute, and cloned iff they are identical or orthogonal (§4.3, proof written for invertible ρ0, ρ1). | Active — read in full 2026-09-25 (NOTE-021); still the most self-contained derivation of mixed-state fidelity as an optimal classical overlap, and of the Holevo bound by the curvature (J” vs S”) argument. Read §3.3 and §3.5.1–3.5.2 for those, and Ch. 2 as a tutorial on why error probability, the Chernoff bound, KL and mutual information are the operationally grounded classical measures. The quantum-Kullback bounds (§3.6), the inference–disturbance formalism (§4.2) and the no-broadcasting proof (§4.3) are preliminary or appeared in companion papers. Read those papers for their final forms. |
| LIT-019 v2 | Why interference phenomena do not capture the essence of quantum theory | Catani et al. (2021), ARXIV-2111.13727. The paper models each mode of the interferometer as a classical bit pair (occupation N ∈ Z_2, phase Φ ∈ Z_2). The agent may know only one of N, Φ or N⊕Φ and must be maximally ignorant of the rest. Under that restriction, local and deterministic dynamics (beamsplitter: swap N_L with Φ_L⊕Φ_R) reproduce exactly the 50-50, ϕ ∈ {0, π} Mach–Zehnder statistics (output port L with probability 1 or 0; ½/½ with a which-way detector). The same holds for the Elitzur–Vaidman bomb tester (success probability ¼), delayed choice and the quantum eraser (interference within each P_A outcome class, none within each Q_A class). So this “TRAP” slice of interference phenomenology cannot, on its own, force wave–particle complementarity, observer-dependence or nonlocal/retrocausal influence. The paper leaves aside the phenomenology it concedes is nonclassical (unbalanced beamsplitters, the functional form of the duality relation, Hardy’s interferometer). | Active — read in full 2026-09-25 (NOTE-018); It is the worked, experiment-by-experiment defence of placing interference in the weakly nonclassical column of LIT-007’s Table II, and it has the clearest statement of the Spekkens-school methodology (define classicality, formalize it, prove a no-go). Read §3, §4.3, §5.1 and App. A. Its noncontextuality claim for the toy field theory is asserted, not proved. |
| LIT-037 v3 | The Open Systems View | Cuffaro & Hartmann (2021), ARXIV-2112.11095. The paper argues philosophically, not formally, that the quantum theory of open systems “GT” is more fundamental than standard closed-system quantum theory “ST”. GT treats states as density operators that evolve, in general, non-unitarily, and it admits not-completely-positive maps. The ontic and epistemic cases each win only on a second, interpretation- or practice-laden criterion: OntFund-2 and EpFund-2. On the first, formal criteria (OntFund-1, EpFund-1) the two frameworks tie, and the authors say so because Stinespring dilation and dropping the Lindblad terms translate in both directions (§4.1, §4.2.1). Explanatory fundamentality is granted only deflationarily (ExFund-1), and the authors recommend dropping the notion (§4.2.2). The step from there to “the open systems view is fundamental tout court” (§4.3) rests on the special sciences and on “there are no other options”. | Active — read in full 2026-09-25 (NOTE-077); worth reading as the clearest statement of the “open systems view” and of the distinction between a view, a framework, a theory and a model (§2, §4.1, §5.2). Read it knowing that its fundamentality verdict is carried by interpretive premises (OntFund-2, EpFund-2) and a deflationary step, not by any formal asymmetry between the frameworks. |
| LIT-045 v5 | Structural Realism | Ladyman (2007), https://plato.stanford.edu/entries/structural-realism/. The entry presents structural realism (Worrall 1989) as the attempt to keep both the no-miracles argument and the lesson of theory change. What is retained, and what we are entitled to believe, is structure, understood by ontic structuralists as modal structure. It sorts the view into an epistemic form, which falls to the Newman/Demopoulos–Friedman triviality result when stated as a Ramsey sentence (§3.2), and an ontic form with seven distinguishable theses, from eliminativism about individuals to “objects are constructs” (§4). The ontic form is motivated by quantum non-individuality and entanglement, gauge symmetry and the hole argument. The entry closes with seven objections, each answered in the author’s own voice (§5). | Active — read in full 2026-09-25 (NOTE-038); the standard map of the structural-realism literature, written by the founder of OSR. Read §2 for the theory-change argument, §3.2 for the Newman problem and §4’s seven OSR theses. Treat §5’s rebuttals as the author’s position, not as the field’s settled view. |
| LIT-053 v2 | The Homological Nature of Entropy | Baudot & Bennequin (2015), DOI-10.3390/e17053253. On a finite probability space, take an information structure S (a set of partitions closed under joins that have a common refinement in S and containing the trivial partition) and a probability family Q complete for it. The information cohomology H*(S;Q) is computed by a Hochschild-type bar complex whose action is Shannon’s averaged conditioning. If S is connected and “sufficiently rich” and cochains are measurable, then H¹ is one-dimensional and spanned by Shannon entropy (Thm 1). This makes entropy the unique measurable 1-cocycle, up to one multiplicative constant per connected component. In the unitary-invariant quantum case H¹_U = 0: Shannon entropy is the coboundary of minus the von Neumann entropy, which is the unique invariant 0-cochain with that property up to an additive constant (Thm 3). The tree (“arborescent”) version again gives H¹ spanned by entropy (Thm 4). | Active — read in full 2026-09-25 (NOTE-088); the founding paper of information cohomology, with complete proofs of the degree-one results (Thms 1, 3, 4). Read it for those. Its broader programme (higher mutual informations as cocycles, super/sub-harmonicity, max-entropy discrimination) is sketched or deferred here to unpublished companion papers. |
| LIT-054 v2 | A complete graphical calculus for Spekkens’ toy bit theory | Backens & Duman (2014), ARXIV-1411.1618. Take the maximal-knowledge fragment of Spekkens’ toy bit theory with post-selected measurements, viewed as relations over {1,2,3,4}ⁿ. It has a ZX-style red–green graphical calculus that is universal (§4.3), sound (§4.4) and complete (Thm 17). Completeness goes through a proof that the toy theory’s binary “check-matrix” formalism coincides with the qubit stabilizer one (valid states SᵀJS = 0, valid transformations QᵀJQ = J, Lemma 3). With it come graph-state/local-complementation results for toy states (Thms 4–5), map–state duality (§5.2) and an rGS-LO normal form (Thms 10–11). The rewrite rules differ from the stabilizer ZX-calculus only through the phase group: Z₂×Z₂ against Z₄. That difference appears as an “11-commutation” rule in place of π-commutation (§4.5). | Active — read in full 2026-09-25 (NOTE-066); the completeness result is proved and still the diagrammatic reference for the toy bit theory. Read §2.1, §4.2, §4.5 and §5.1 for the stabilizer comparison. The ψ-epistemic framing in the abstract and §1 is motivation only; the paper does no ψ-epistemic analysis. |
| LIT-062 v3 | On the reality of the quantum state | Pusey et al. (2011), ARXIV-1111.3328. Take pure states |ψ0⟩, |ψ1⟩ with |⟨ψ0|ψ1⟩| = cos θ. Assume (a) a model assigns each preparation a distribution µ_i(λ) over physical states, (b) n independent preparations yield the product distribution µ_{x1}(λ1)⋯µ_{xn}(λn), and (c) outcome probabilities depend only on (λ1, …, λn). If the model’s predictions are within ε of quantum theory, then D(µ0, µ1) ≥ 1 − 2ε^{1/n} for any n with 2^{1/n} − 1 ≤ tan(θ/2) (Eq. 7, App. B). At ε = 0 the distributions for distinct pure states cannot overlap on a set of non-zero measure, so Harrigan–Spekkens ψ-epistemic models are ruled out. | Active — read in full 2026-09-25 (NOTE-030); the founding ψ-ontology theorem. It is short, fully proved (App. A–B), and still the reference that every later ψ-epistemic/ψ-ontic argument, including LIT-007 §I.C and LIT-090, positions itself against. Read it with its assumption (b)–(c) in view, since that is where every response pushes. |
| LIT-073 v2 | General probabilistic theories: An introduction | Plávala (2021), ARXIV-2103.07469. A self-contained textbook on finite-dimensional GPTs whose central results are that state spaces and effects are dual (K = {φ ∈ A(K)*+ : ⟨φ, 1_K⟩ = 1}, Thm 3.19) and that optimal two-state discrimination succeeds with probability ½(1 + ‖λx₀ − (1−λ)x₁‖) in base norm (Thm 3.43). Every composite lies between the minimal and maximal tensor products (Prop. 5.6), and four statements are equivalent: id_K is measure-and-prepare, id_K is self-compatible, K admits a universal broadcasting channel, and K is a simplex (Thm 7.7); minimal = maximal iff one factor is a simplex is cited to [134], not proved. | Active — read in full 2026-09-25 (NOTE-035); the best single entry point to the GPT vocabulary this record’s Spekkens-programme readings presuppose (state space, effect algebra, GPT dimension, tomographic locality, minimal/maximal tensor products, measure-and-prepare). Read §2–3, §5.1, §5.5, §6.1–6.5 and §7.1. It is not the place for restricted theories like Spekkens’ toy theory, to which §3.7 gives a single paragraph. |
| LIT-077 v2 | What is quantum in quantum randomness? | Grangier & Auffèves (2018), ARXIV-1804.04807. The paper argues informally, with no equations anywhere in it, that an “elementary” system with a fixed finite number N of exclusive modalities, placed in contexts that can change, must behave unpredictably when the context changes: a return to C1 after C2 cannot be certain, or there would be N² > N joint modalities (p. 4). It asserts that adding one condition, that the context can change continuously, makes the N×N modality-transition matrix unistochastic, i.e. Born-rule, but defers that proof to refs [14, 15] (p. 5). The anti-reductionist conclusion follows from Postulate 0, which declares contexts non-contextual by definition, not from an argument against decoherence. | Active — read in full 2026-09-25 (NOTE-044); a short, equation-free position paper whose one argument of its own is an informal N = 2 reductio; read it for the position, and read the derivation it cites — Auffèves & Grangier, Sci. Rep. 2017 (arXiv 1610.06164) and arXiv 1801.01398, neither held here — for the result. |
| LIT-081 v2 | Ensemble Steering, Weak Self-Duality, and the Structure of Probabilistic Theories | Barnum et al. (2009), ARXIV-0912.5532. A bipartite state ω steers its B-marginal iff ω̂ : [0,u_A] → [0,ω_B] is a strong quotient map of ordered sets (Thm 5.4). When ω̂ is injective and the marginal is interior, a steering state is an order-isomorphism (Cor. 5.5). With Thm 4.1/Cor. 4.2, it follows that universal self-steering makes every irreducible state space homogeneous and weakly self-dual (Prop. 5.8). The step from there to C*-algebraic quantum theory needs strong self-duality, which the paper does not derive, and a Hanche-Olsen correspondence it defers (“as we will establish elsewhere”, §6). | Active — read in full 2026-09-25 (NOTE-082); a short, readable and mostly proved link from an operational feature (ensemble steering) to cone geometry (homogeneity, weak self-duality). It remains a standard reference for the steering route in GPT reconstructions. Read it with the open weak/strong self-duality gap and the thin argument for Prop. 5.7 in view. |
| LIT-082 v2 | Unification of observational entropy with maximum entropy principles | Schindler et al. (2025), ARXIV-2503.15612. The paper defines S^τ_M(ρ) = S(τ) − D_M(ρ‖τ) = −Σ_x p_x log(p_x/V_x), with V_x = Tr(τM_x)e^{S(τ)} (Eqs. 10–12). It is bounded as S(τ) ≥ S^τ_M(ρ) ≥ S(ρ) when ρ obeys the constraint S(ρ;τ) ≤ S(τ) (Eqs. 17–18). It reduces to 24 listed entropies, and it removes traditional observational entropy’s infinite-dimensional pathologies whenever S(τ) < ∞ (§VIII.7). Its “second laws” are bounds D_M(ρ̄‖τ) ≤ ε on the time-averaged state, split as equilibration plus prior-fit (Eq. 27). The paper proves the first term small in two regimes (Eqs. 28–29) and the second only under the ETH (Eq. 31). It states that in its own numerical examples the smallness “was [not] proved here analytically” (p. 12). | Active — read in full 2026-09-25 (NOTE-043); worth reading as the reference definition of prior-generalised observational entropy and for its catalogue of special cases (§V, Table II). The unification is definitional and its bounds are correct. The second-law programme is a framework plus partial sufficient conditions, not a proved second law for any concrete system. |
| LIT-086 v2 | The Stochastic-Quantum Correspondence | Barandes (2023), ARXIV-2302.10778. Take any N×N stochastic transition matrix Γ(t←0) on a configuration space. Writing it entrywise as Γ_ij = |Θ_ij|² = tr(Θ†P_iΘP_j) (Eq. 15) is always possible, and the paper calls it the stochastic-quantum dictionary. Choosing Θ unitary (by Stinespring dilation to ≤N³ configurations, §3.4) makes Γ unistochastic. The Born rule in the configuration basis (Eq. 23), the Schrödinger equation (Eq. 35, with H ≡ iħ(∂U/∂t)U†) and interference (Eq. 43, the gap between Γ(t←0) and |U(t←t′)|²|U(t′←0)|²) then follow by definition or short algebra. The Born rule for non-diagonal observables (Eq. 78) is derived by reading a measuring device’s configuration, given a composite unitary that produces Eq. (73). | Active — read in full 2026-09-25 (NOTE-080); the clearest single statement of the indivisible-stochastic-process reading of quantum theory, and its measurement model (§4) is worth reading. Discount its “new result” labels: the dictionary is an identity, the Schrödinger equation is the definition of H, and unitarity is a representational choice the paper’s own gauge footnote does not preserve. |
| LIT-090 v3 | Could wavefunctions simultaneously represent knowledge and reality? | Hance et al. (2021), ARXIV-2101.06436. The paper argues, informally and without formal results, that Harrigan–Spekkens’ ψ-ontic/ψ-epistemic are contradictories only by stipulation. On the informal senses, “represents reality” and “represents knowledge”, a wavefunction can be both. It offers a weaker sufficient condition for “epistemic” (Eq. 6: for some ψ, A, S, the response function A(S|λ) is not equal to the Born probability p^ψ_A(S) throughout Λψ), which non-overlapping models can meet. It concludes that PBR “shows that wavefunctions cannot overlap on state space, but it says nothing about whether they are epistemic in the broader sense”, so it “does not rule out all epistemic models” (§V). | Active — read in full 2026-09-25 (NOTE-072); a short, clear statement that ψ-epistemic in the Harrigan–Spekkens sense is narrower than “epistemic”, worth reading for §III–V as a terminological caution when citing PBR. Its positive criterion (Eq. 6) is weak, since it amounts to predictive incompleteness of ψ. Its Appendix A contains claims that the record’s own readings contradict. |
| LIT-100 v2 | Quantum States for Primitive Ontologists: A Case Study | Belot (2011), DOI-10.1007/s13194-011-0024-8. Inside primitive-ontology Bohmian mechanics, Belot compares four readings of the quantum state: a field on configuration space, a multi-field on 3-space, a law, and a history of velocity dispositions of the particle system. Every reading except the multi-field carries a hard open question: why the world looks 3-dimensional, whether laws can be time-indexed, and whether dispositions lose content. The last is shown by the particle in a box, where every energy eigenstate φn = sin(πnx) induces the same at-rest disposition. The orthodox should prefer the multi-field reading; Belot, on van Fraassen’s view of interpretation, prefers to keep the others alive. | Active — read in full 2026-09-26 (NOTE-110); worth reading. It is short, it originated the dispositionalist reading, and the particle-in-a-box counterexample and the analysis of the costs of ψ-as-law are still the points the later literature (LIT-204, LIT-200, LIT-164) cites. |
| LIT-107 v3 | Quasi-set theory: a formal approach to a quantum ontology of properties | Holik et al. (2021), DOI-10.1007/s11229-022-03884-8. The paper argues, with no new theorem, that the Lombardi–da Costa ontology of properties for quantum mechanics can be stated in Krause’s quasi-set theory Q. On that ontology a system is a bundle of I-type-properties (instances of universal type-properties, represented by the algebra of observables) with no principle of individuality. For each property instance, the indiscernible instances form a qset U([Π]). An atomic bundle is any qset x with qc(x ∩ U) = 1 for each U; the Axiom of Choice in Q guarantees one exists. The bundle’s kind is Q(B) = {x ∈ P(Q) | R(x)}, whose members are all indiscernible. N particles are a subqset N(B) ⊂ Q(B) with qc = N, a definite number with no labels (§6, pp. 19–22). | Active — read in full 2026-09-26 (NOTE-127); worth reading as a short, clear bridge between the modal-Hamiltonian “ontology of properties” (Lombardi et al.) and quasi-set theory (French & Krause 2006). The new formal content is about two pages (§6). Most of §5 restates Q’s axioms, and the printed proofs contain errors. It is a useful statement of the position, not a result. |
| LIT-113 v3 | Duality and ontology | Le Bihan & Read (2018), DOI-10.1111/phc3.12555. A survey that sorts realist responses to dualities along two axes. The first is which of the naively interpreted dual worlds count as candidates for actuality: all of them (underdetermination), one (discrimination) or none (nihilism), each with or without the common-core world. That gives options (1)–(6), with the common-core approach being (1), (3) and (5). The second axis is what replaces the common core: an overarching theory, or “pluralism”, on which all dual structures are co-instantiated as parts of one world. Its verdicts are conditional: no option breaks the underdetermination without a further metaphysical principle, such as the PII, for which the authors see no independent support. | Active — read in full 2026-09-26 (NOTE-152); worth reading as the shortest clear map of the realist options for interpreting dualities. It is a nontechnical survey, not an argument for any option. Its original contributions (the discrimination/common-core overlap, a defence of discrimination for AdS/CFT, and pluralism) are presented as options, not established. |
| LIT-121 v2 | The Meaning of the Wave Function: In Search of the Ontology of Quantum Mechanics | Gao (2016), ARXIV-1611.02738. Gao argues first that ψ is real. His case uses protective measurements, which yield ⟨A⟩ on a single system with certainty, in three versions (ch. 4): one in the ontological-models framework using preparation non-contextuality, two he says need no auxiliary assumption, and one that uses a new criterion of reality. He then argues that ψ describes random discontinuous motion (RDM) of N point particles in 3-space: |ψ|² is the density with which each particle is present at each instant, and ψ is a single-case propensity. The main support is that the charge density |ψ|²Q, which protective measurement shows to exist, has no self-interaction (chs. 6–7). If the Born probabilities come from RDM, collapse is real. Gao proposes an energy-conserving, Planck-time-discrete collapse model: Pᵢ(t+t_P) = Pᵢ(t) + (ΔE/E_P)[δ_{EsEi} − Pᵢ(t)], giving τ_c ≈ ħE_P/(ΔE)² (ch. 8). Combined with relativity, it requires a detectable preferred Lorentz frame (ch. 9). | Active — read in full 2026-09-26 (NOTE-140); worth reading as the fullest single statement of the protective-measurement argument for ψ-reality and of the RDM ontology. Its chapters 2, 3, 7.1 and 8.5 are also a well-referenced critical survey of ψ-epistemic, nomological, configuration-space-realist, Penrose and CSL views. Its central premises are contested (see Limitations), its collapse model is unconfirmed, and chapters 8–9 are openly speculative. |
| LIT-123 v2 | Reality as a Vector in Hilbert Space | Carroll (2021), ARXIV-2103.09780. A programmatic essay, not a result paper. Carroll argues that a quantum theory is fixed, up to unitary equivalence, by the Hamiltonian’s eigenvalue list {E_n} on a finite-dimensional factor (he assumes a non-degenerate spectrum), and that parts are not given. Subsystems (tensor factorisations H = H_S ⊗ H_E or ⨂_α H_α), classicality and space are to be selected afterwards as rare structure that suits the dynamics. The only uniqueness he reports is for local factorisations, and it rests on a cited theorem (Cotler et al. 2019). The system/environment criterion rests on “simple numerical examples” in a companion paper, the metric and Einstein’s equation on “optimistic assumptions”, and emergent fields on a hope. | Active — read in full 2026-09-26 (NOTE-134); worth reading as the shortest authoritative statement of Hilbert-space fundamentalism and its “quantum mereology” programme, with the caution that nothing is derived in it. Every technical claim is delegated to a companion paper, and the author says the programme “may ultimately fail” (p. 12). |
| LIT-133 v2 | Towards a Generalized Theory of Observers | Elshatlawy et al. (2025), ARXIV-2504.16225. The paper defines a “minimal observer” as a finite-state transducer with a boundary: a tuple O = (X, Y, Z, f, g, B) with |X| > 1, |Y| ≥ 1, |Z| ≥ 1 and feedback closure (Def. 4.1, p. 57). It then asserts, without derivation, that measurement outcomes, reference frames, coarse-graining and the quantum–classical cut are all defined by such observers. The paper’s only formal results are that isomorphism of these machines is an equivalence relation (“Theorem 2”, p. 93), and a complexity bound C(O) ≥ log 2 (Prop. 4.1, p. 94) whose correction term Λ is never defined and whose proof supports only C(O) ≥ 0. | Rejected — read in full 2026-09-26 (NOTE-121); not worth a reader’s time as a theory of observers. Its one definition is a Moore machine with an unformalised boundary, its “theorems” are definitional, and its quantum claims are asserted, including the claim to identify “the exact moment and mechanism of quantum collapse” (p. 75). A reader wanting minimal observers or agents is better served by the sources it surveys: Ashby, Hartle’s IGUS, Fields, and Gielen & Wise’s observer space in §4.10.3. |
| LIT-164 v3 | Wave-Function Realism and the Scope of Physical Causal Closure | Poltorak (2026), https://philsci-archive.pitt.edu/30720/. A closure principle has two tacit parameters, its domain and its completeness relation (causal, dynamical or explanatory). Wave-function realism falsifies spatiotemporal-domain closure (STC) but is compatible with fundamental-physical closure (FPC), so there is no “trans-domain causal breach”. The physicalist is left with a dilemma: STC is too narrow, and FPC needs an independent criterion of physicality plus a dynamical-to-causal bridge. The paper tests this across primitive-ontology, spacetime-state, multi-field, Everettian, nomological and dispositional readings. | Active — read in full 2026-09-26 (NOTE-112); worth reading as the clearest statement of what quantum ontology does and does not do to causal-closure arguments. Its conclusions are modest, and it concedes that its main target is a picture rather than a published argument. Unrefereed. |
| LIT-170 v2 | Unitary inequivalence as a problem for structural realism | French (2010), DOI-10.1016/j.shpsb.2011.09.001. French answers Howard’s objection that QFT’s unitarily inequivalent representations leave the structural realist with non-isomorphic, competing structures. He offers two routes. The first is to adopt cut-off “conventional” QFT with Wallace, on which the inequivalence is an artefact or harmless. The second is to follow Ruetsche’s Swiss Army Knife and treat the algebra as a modal structure spanning all representations. On that reading spontaneous symmetry breaking preserves the symmetry across the degenerate vacua, and the DHR analysis derives quantum statistics from the algebra. Adding Baker’s criterion, on which a symmetry mapping between representations suffices for intertranslatability, removes the assumption behind the objection, and French ends with a third option that combines the two routes. Every step is an informal argument layered on others’ technical results; the paper proves nothing itself. | Active — read in full 2026-09-26 (NOTE-162); worth reading as a leading ontic structural realist’s worked response to the strongest physics-based objection to OSR beyond finite-dimensional QM, and for its clear staging of the Wallace–Fraser and algebraic versus Hilbert-space debates. A reader should cite the 2012 journal version; this note reads the 2010 draft. |
| LIT-173 v2 | From ontic structural realism to metaphysical coherentism | Morganti (2018), DOI-10.1007/s13194-018-0233-5. Morganti argues from a survey rather than proving anything. Strong, moderate and contextualist OSR each fail as metaphysics: particle kinds have intrinsic features no group-theoretic analysis reduces, and the argument from permutation invariance is inconclusive. What OSR gets right about symmetry, entanglement and particle kinds is better captured by “metaphysical coherentism”: object-like entities (in one domain) stand in symmetric partial ontological dependence on one another. Cycles of such dependence can be kept irreflexive, and so explanatorily non-trivial, by relaxing transitivity (quasi-transitivity) or by making only partial dependence symmetric. | Active — read in full 2026-09-26 (NOTE-095); worth reading as the clearest statement of a dependence-based, object-friendly alternative to relation-first OSR, and a hub for the grounding, holism and OSR literatures; its physics applications are programmatic, and he says so himself (pp. 18–19). |
| LIT-176 v2 | Events and the Ontology of Quantum Mechanics | Dorato (2015), ARXIV-1503.01140. Dorato argues, without formal results, that events are the metaphysical category best suited to quantum mechanics. An event reading fits GRW flashes, GRW matter density and Bohmian worldlines, and also the antirealist views of ψ held by Bohr, Rovelli, Haag and Pashby. Because it is the same across interpretations, he calls it “interpretation-independent” (p. 12). Where ψ needs a realist reading, it should be read as intrinsic dispositions of events. Esfeld et al.’s view has the opposite structure: bare matter points plus a single holistic power of the universal configuration. Dorato argues that view is unintelligible, and that it must change its ontology when it passes from the universal wave function to the effective one (pp. 13–15). | Active — read in full 2026-09-26 (NOTE-105); worth reading as a short, clear statement of event-primitivism for primitive-ontology theories, and of intrinsic (not holistic) dispositions against Esfeld et al. It is programmatic. By Dorato’s own account (p. 15) the relation between events and powers is left for future work, and without that the event reading does little work beyond relabelling. |
| LIT-177 v2 | The Relational Interpretation of Quantum Physics | Rovelli (2021), ARXIV-2109.09170. Rovelli restates RQM as an ontology of sparse relative facts, which arise in interactions and are labelled by the interacting systems. Its one technical rule is that an amplitude W(b,a) gives the probability P = |W|² only when a and b are relative to the same system. That rule removes the clash between P_collapse = Σᵢ|W(c,bᵢ)|²|W(bᵢ,a)|² and P_unitary = |ΣᵢW(c,bᵢ)W(bᵢ,a)|² (eqs. 2–4). Decoherence makes some facts approximately label-independent (“stable”). ψ is a relative-state bookkeeping device, like a Hamilton–Jacobi function S. The price, which Rovelli states, is giving up “strong realism”: values are not held at all times, may be held at different times for different systems, and there is no coherent global view and no quantum state of the universe. | Active — read in full 2026-09-26 (NOTE-174); worth reading as the author’s own compact, current statement of RQM, and a good anchor citation for relational and perspectival readings. It is expository, not argumentative. It does not engage the inter-observer consistency objections in any depth, and the main technical rule is postulated, not derived. |
| LIT-184 v2 | Effective Ontic Structural Realism | Ladyman & Lorenzetti (2023), DOI-10.1086/729061. Ladyman and Lorenzetti argue that three recent “effective realisms” lack a metaphysics that makes them realist rather than constructive-empiricist, and that the Ladyman–Ross OSR package supplies one: realism about modal structure, “to be is to be a real pattern”, and scale-relative ontology. The three are Fraser & Vickers (and Egg) on non-relativistic QM, Williams and Fraser on QFT, and Robertson & Wilson on “theoretical relicts”. Scale relativity also dissolves Saatsi’s inconsistency objection, so Newtonian gravitational forces and GR gravitation “both exist at different scales”. Egg’s 2019 anthropocentrism objection is answered by grounding measurement in decoherence. | Active — read in full 2026-09-26 (NOTE-129); worth reading as a short, clear map tying the effective-realism literature to OSR’s modal-structure and real-patterns ontology; its arguments are programmatic, and each step leans on Ladyman & Ross (2007) rather than new argument. |
| LIT-199 v3 | The Philosophy and Physics of Duality | De Haro & Butterfield (2025), ARXIV-2508.01616. The book gives a formal Schema: a duality is an isomorphism between the “model roots” (state space, quantities, dynamics) of two models of a common-core bare theory, with values and dynamics matching. It pairs this with an interpretative condition: duals are physically equivalent iff (A) they are duals and (B) i1 = i2∘d. On that basis it argues four things. First, there are no genuine cases of in-principle underdetermination in the examples, and a cautious realist can commit to the common core. Second, dualities do not favour structural realism. Third, duality is formally incompatible with ontological emergence, except “beside” it (AdS/CFT’s RG direction), via effective dualities, or epistemically. Fourth, quasi-dualities motivate a “geometric view” on which models are charts on a successor theory’s moduli space. | Active — read in full 2026-09-26 (NOTE-178); worth reading as the reference treatment of dualities in philosophy of physics. It supplies definitions (the Schema, §3.1, and physical equivalence, §12.1.2, p. 402) and a common vocabulary (internal/external interpretation, common core, quasi-duality) that the rest of the record’s duality works either use or argue against. Its physics examples are admittedly partly conjectural (AdS/CFT, Montonen–Olive and M-theory are unproven), and its realism is a stated position, not a derivation. |
| LIT-200 v2 | Wave-functionalism | Allori (2021), DOI-10.1007/s11229-021-03332-z. Allori proposes that the wave function be defined by three roles — empirical, nomological and explanatory. On this view it is an ingredient that “partially realizes” the law, not a law and not a material field. So its time-dependence and non-uniqueness (a phase shift, a density matrix) are harmless, it counts as non-epistemic and ontic, and it can take part in constructive explanation. Each of these is argued informally, and the escape from PBR is by relabelling the ontic/epistemic distinction, not by engaging the theorem. | Proposed — read in full 2026-09-26 (NOTE-154); a distinct and clearly placed option in the ψ-as-law family (LIT-204 §4.2). It is unproven until “part of the law” and “non-representational yet ontic” are given content that separates the view from Humean nomologism, and until it answers the “too cheap” objection it invites. |
| LIT-204 v2 | Realism about the Wave Function | Chen (2018), ARXIV-1810.07010. A survey that sorts realist readings of the non-relativistic universal wave function into four ontological readings (a field on 3N-dimensional space, a multi-field on 3-space, properties of systems as in spacetime state realism, a Hilbert-space vector), two nomological ones (strong: time-independent and simple like the classical Hamiltonian; weak: a Humean best-system variable) and Maudlin’s sui generis option. It tabulates their compatibility with BM, GRW0/m/f and S0/Sm and with high- versus low-dimensional fundamental space (Table 1, 10 of 56 cells ”?”), and declares no winner. | Active — read in full 2026-09-26 (NOTE-175); worth reading as the compact map of the realist options and the hub for the wave-function cluster (LIT-200, LIT-100, LIT-164, LIT-121, LIT-176), with the caution that Table 1 is asserted, not derived, and is uneven in places. |
| LIT-217 v3 | Science, Metaphysics and Structural Realism | Ladyman (2001), DOI-10.21825/philosophica.82261. Ladyman argues that structural realism should be metaphysically, not epistemically, revisionary. He gives three grounds. Quantum statistics leaves the individuality of particles underdetermined: permutation invariance gives 3 two-particle states, not 4, so probabilities are 1/3 rather than 1/4 (p. 62). Entangled states give non-supervenient relations that refute Humean supervenience (pp. 71–73). The hole argument blocks point-based substantivalism (pp. 68–69). He proposes that science describes “the modal structure of the phenomena” (p. 73), with modal realism supplying the no-miracles intuition. The paper ends non-eliminatively: “the phenomena have structure but they are not structure” (p. 74). | Active — read in full 2026-09-26 (NOTE-192); Worth reading as the earliest freely available statement of ontic structural realism by its originator. It argues the Humean-supervenience and quantum-individuality points more fully than the SEP survey (LIT-045). It also shows that the 2001 view was a modest one, realism about modal structure with no eliminativism, before Every Thing Must Go. Its positive proposal takes two paragraphs (pp. 73–74) and is programmatic. |
| LIT-237 v2 | Bra–ket notation (Wikipedia) | Wikipedia contributors (2026), https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation. Dirac’s bra–ket notation writes vectors as kets and continuous linear functionals as bras, and on a Hilbert space the Riesz representation theorem is what licenses treating every bra as the conjugate of a ket, so that ⟨φ|ψ⟩ is at once a functional applied to a vector and an inner product. | Active — read 2026-10-14 (NOTE-855) in full; worth reading for the conventions of Dirac notation and for its plain statement that the bracket is an inner product only where the Riesz theorem holds. Read it knowing that it is Wikipedia under a cleanup banner, that its lead was edited on 2026-09-30 to date the notation to a 1930 monograph while its own reference and later text say 1939, that its pointer to GNS for non-normalizable kets is weak, and that two of its sections contradict each other on |ψ⟩† = ⟨ψ|. |
| LIT-241 v2 | Gelfand–Naimark–Segal construction (Wikipedia) | Wikipedia contributors (2026), https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction. Any state (positive normalized linear functional) on a C*-algebra determines, essentially uniquely, a Hilbert space and a cyclic *-representation in which that state becomes a vector expectation ρ(a)=⟨π(a)ξ,ξ⟩, and the representation is irreducible exactly when the state is pure. | Active — read 2026-10-14 (NOTE-857) in full; worth reading as a short reference statement of the GNS construction: a state on a C*-algebra gives a cyclic representation, unique up to the unitary π(a)ξ ↦ π′(a)ξ′, irreducible exactly when the state is pure. Read it knowing that it is Wikipedia, that the uniqueness is stated with a reference to Kadison rather than proved, and that it is about states on complex C*-algebras, not kernels on sets: the kernel form the record uses comes from LIT-259. |
| LIT-259 v2 | Noncommutative analysis, Multivariable spectral theory for operators in Hilbert space, Probability, and Unitary Representations | Jorgensen & Tian (2014), ARXIV-1408.1164. The textbook calls building a Hilbert space from a positive definite function “the GNS construction” (Remark 1.34) and states (Cor. 1.35), leaving the proof as an exercise, that the feature map realising a kernel as inner products is unique up to a unitary among minimal realisations. It proves the parallel uniqueness for states and cyclic representations (Thm 4.35), says that in the abelian case states are probability measures, so that “the GNS construction is non-commutative measure theory” (Ex. 4.28), and writes the commutative GNS space out as L²(μ) (Ex. 4.86). | Active — read 2026-10-14 (NOTE-829) in part; worth reading for the passages that put the kernel construction and the GNS construction under one name: Remark 1.34 builds a Hilbert space from a positive definite function by the GNS recipe, Corollary 1.35 states that minimal realizations ϕ(x,y) = ⟨Φ(x), Φ(y)⟩ are unique up to a unitary, Theorem 4.35 proves the GNS theorem with its uniqueness, and Example 4.86 writes the commutative GNS space out as L²(μ). Read it knowing that Corollary 1.35 is not proved (the proof is “left as an exercise”; the argument of Theorem 4.35 transfers), that equation (1.24) carries a typo, that “GNS” for a kernel on a set is a naming and not a stated reduction, and that only the cited passages of this 442-page text were read. |
| LIT-262 v2 | The Geometry of Information Retrieval | van Rijsbergen (2004), DOI-10.1017/CBO9780511543333. The book recasts IR’s vector-space, probabilistic and logical models in one Hilbert-space language. Documents are state vectors. Relevance and aboutness are Hermitian observables that need not commute. Subspaces carry a non-distributive logic whose conditional is shown to be a Stalnaker conditional equal to the Sasaki hook (ch. 5). Gleason’s theorem makes tr(ρP) the probability of a subspace, which ch. 6 uses to rewrite cosine matching, cluster representatives, relevance and pseudo-relevance feedback, dynamic clustering and ostensive retrieval. It is a proposed language, not a tested model: no experiment is reported, and its “content hypothesis” that the probability of a concept is cos²θ is argued, not tested. | Active — read in full 2026-09-27 (NOTE-239); worth reading as the founding statement of the Hilbert-space (quantum-probability) programme in IR. It is short, self-contained and mathematically careful at the level it works at, and its ch. 5 proof that the subspace conditional is a Stalnaker conditional and its ch. 6 trace-formula recastings of standard IR operations are the parts to take away. Read it as a language, not as evidence: it reports no experiment, and every IR payoff it names is a reformulation or a research direction. |
| LIT-263 v2 | Kochen-Specker contextuality | Budroni et al. (2021), ARXIV-2102.13036. A review that fixes a minimal definition of Kochen–Specker contextuality: no global joint distribution has the context distributions as marginals (Fine’s theorem). The sheaf-theoretic framework is one of several formulations it calls substantially equivalent. It argues that contexts defined by joint measurability alone make every nondisturbing correlation classically realizable, so contexts must be repeatable, nondisturbing measurements. It treats Spekkens’s generalized contextuality as a different notion, covered only to mark the difference. | Active — read 2026-10-14 (NOTE-844) in full; worth reading as the record’s reference for Kochen–Specker contextuality: the marginal-problem definition (Fine’s theorem) with the sheaf form among its equivalents, why contexts need more than joint measurability, how experiments model disturbance, and Vorob’ev’s acyclicity theorem. Read it knowing that it is a review whose results are mostly cited, that it describes Spekkens’s notion only to set it apart and never derives the relation between the two, and that its account of maximally noncontextual models is one noise model among several, with no behavioural data. |
| LIT-265 v2 | The contextual fraction as a measure of contextuality | Abramsky, Barbosa & Mansfield (2017), Phys. Rev. Lett. 119, 050504. For any no-signalling empirical model in the sheaf-theoretic framework, the contextual fraction CF (one minus the largest weight of a noncontextual sub-model) is a linear programme. Its dual gives a generalised Bell inequality whose normalised violation equals CF, the largest attainable (Theorem 1). CF does not increase under relabelling, translation of measurements, coarse-graining or mixing, and is multiplicative in NCF under products (Theorem 2). It lower-bounds the failure probability of Z2-linear MBQC computing a non-linear function (Theorem 3). | Active — read 2026-10-09 (NOTE-236); worth reading as the reference definition of the contextual fraction CF. CF is the weight left over when the largest noncontextual sub-model is taken out of a no-signalling empirical model. It is a linear programme over the incidence matrix (Eq. 3). Its dual reads off a Bell inequality whose normalised violation is exactly CF, the maximum over all Bell inequalities (Theorem 1). CF is invariant under relabelling, cannot increase under translation of measurements, coarse-graining of outcomes or mixing, and composes as max under choice and as 1 − NCF·NCF′ under product (Theorem 2). It lower-bounds the failure probability of Z2-linear MBQC and of k-consistent games (Theorems 3–4). All proofs are complete in the supplement. It is defined only for models with compatible marginals. |
| LIT-266 v2 | Contextuality for preparations, transformations, and unsharp measurements | Spekkens (2004), arXiv:quant-ph/0406166. Defines a noncontextual ontological model as one in which operationally equivalent procedures (preparations, measurements, transformations) get identical representations; shows the traditional Kochen–Specker notion is the special case of measurement noncontextuality for sharp measurements plus outcome determinism, proves that preparation noncontextuality implies that outcome determinism, and proves preparation contextuality already for a qubit. | Active — read 2026-10-14 (NOTE-748) in full; worth reading as the source of generalized noncontextuality and as the bridge between it and the Kochen–Specker notion. It proves preparation, transformation and unsharp-measurement contextuality for a qubit, shows the traditional notion is measurement noncontextuality plus outcome determinism for sharp measurements, and proves that preparation noncontextuality implies that determinism. Read it knowing that every result assumes exact operational equivalence and gives no inequality; that the bridge (§VIII.A) uses the full set of quantum preparations; and that “any classical theory is necessarily noncontextual” is a footnote assertion. It argues from ontological economy and does not mention Leibniz. |
| LIT-274 v2 | A new description of orthogonal bases | Coecke et al. (2008), ARXIV-0810.0812. Theorem 5.1 proves that on a finite-dimensional complex Hilbert space, commutative †-Frobenius monoids in FdHilb and orthogonal bases are in bijection: the basis is the monoid’s copyable elements, and the monoid is the linear extension of copying and uniformly deleting that basis. §6 adds that the monoid is special (m∘δ = id) exactly when the basis is orthonormal. Corollaries 7.1 and 7.2 lift this to categories: with fully structure-preserving maps the category is equivalent to the groupoid of finite sets labelled by positive reals (the basis norms), and with comonoid homomorphisms it is equivalent to FinSet. | Active — read in full 2026-09-27 (NOTE-247); worth reading as the short, self-contained source of the fact that “a commutative †-Frobenius algebra” and “an orthogonal basis” are the same data in FdHilb, and “special” the same as “orthonormal”. The proof is complete in outline, and the heavy step is delegated to the spectral theorem for finite-dimensional commutative C*-algebras. Read §3’s displayed formulas (3) and (4) as the orthonormal case only, and the result as complex and finite-dimensional only. |
| LIT-275 v2 | H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics | Abramsky & Heunen (2010), ARXIV-1011.6123. In the category Hilb of complex Hilbert spaces of arbitrary dimension, a Frobenius algebra with a unit exists only in finite dimension (Lemma 3). So the paper drops the unit, and Theorem 22 proves that a nonunital commutative special †-Frobenius algebra corresponds to an orthonormal basis (the basis being the algebra’s copyable vectors) if and only if it is semisimple, if and only if it satisfies Ambrose’s H*-axiom (H), and if and only if it is a directed colimit of finite-dimensional unital ones. Whether (H) is automatic, i.e. whether a nonzero “radical” Frobenius algebra exists (Proposition 23), is left open. In categories of relations and of nonnegative ℓ²-matrices, Frobenius algebras decompose as disjoint unions of abelian groups and do satisfy (H) (Theorems 30–38). | Active — read in full 2026-09-27 (NOTE-248); worth reading as the precise statement of how far “a basis is a Frobenius algebra” survives in infinite dimensions. It gives the unit’s failure, the H*-algebra substitute and five equivalent conditions, all proved or reduced to Ambrose (1945), with one question left open. Read “arbitrary bases” in the abstract as “orthonormal bases of arbitrary dimension over ℂ”, and “observables” as discrete-spectrum observables only. |
| LIT-276 v2 | Measurement as Sheafification: Context, Logic, and Truth after Quantum Mechanics | Ghose (2025), ARXIV-2512.12249. A programmatic essay, with no theorem, model or computation of its own. It restates Abramsky & Brandenburger’s criterion (contextuality and non-locality as the absence of a global section of a presheaf over measurement contexts) and adds three things. It declares measurement to be “sheafification” of that presheaf, without specifying a site, a topology or any sheafified object. It asserts that the Ghose–Patra seven-valued logic is a finite Heyting algebra, without giving an order or operations. It proposes that a quantum potential scaled by 0 ≤ λ ≤ 1 (λ = 1 quantum, λ = 0 classical Hamilton–Jacobi) carries the presheaf continuously to a sheaf, which is asserted, not modelled. Its one worked formula, v2’s Čech “obstruction class” [δs], is a coboundary and so is zero by construction. | Rejected — read in full 2026-09-27 (NOTE-249); not worth a reader’s time for its formal claims. Every correct formal ingredient is Abramsky & Brandenburger’s (LIT-016) or Abramsky, Mansfield & Barbosa’s, and both are better read in the original. What the essay adds is either unconstructed (the sheafification, the site, the seven-element Heyting algebra, the link from λ to cohomology) or wrong as stated: v2’s obstruction class is zero by construction, a sheaf topos is not in general Boolean, and classical distributions do not form a sheaf. Its only value is as a record of a position, the Bohrian reading of contextuality plus the Jaina analogy. |
| LIT-277 v2 | The Cohomology of Non-Locality and Contextuality | Abramsky et al. (2011), ARXIV-1111.3620. For an empirical model e on a measurement cover 𝒰, a section s in the support of context C₁ gets a class γ(s) ∈ Ȟ¹(𝒰, F_{C̄₁}), relative Čech cohomology of the presheaf F = F_ℤ S_e of formal ℤ-linear combinations of support sections. γ(s) = 0 iff s belongs to a compatible family of such ℤ-combinations (Prop 4.2), so a non-vanishing γ(s) is a sufficient but not necessary witness of contextuality (Prop 4.3). γ is non-zero for every section of the PR box, GHZ, the triangle, the 18-vector Kochen–Specker set and the Peres–Mermin square, but it vanishes on the Hardy model and on one strongly contextual Kochen–Specker cover (§8). | Active — read in full 2026-09-27 (NOTE-250); worth reading as the short original source of the cohomological obstruction to contextuality. It gives the exact definition (a relative Čech class, with coefficients the free abelian group on support sections), the vanishing criterion and its two known failures. Its results are small propositions with complete proofs, plus machine-checked mod-2 computations that I reproduced. It calls itself “work in progress” (§1), and the conceptual development is in Abramsky et al.’s CSL 2015 “Contextuality, Cohomology and Paradox” (not read). A reading of that paper would show whether it supersedes this one. |
| LIT-278 v2 | Contextuality, Cohomology and Paradox | Abramsky et al. (2015), ARXIV-1502.03097. For a model whose outcomes live in a commutative ring R, “All-vs-Nothing” (AvN_R) means the R-linear equations satisfied by every support section of every context have no global solution. Theorem 21 proves AvN_R(S) ⇒ SC(Aff S) ⇒ CSC_R(S) ⇒ CSC_ℤ(S) ⇒ SC(S), so every AvN model is strongly contextual with a non-vanishing Čech obstruction on every section. Theorem 4 shows that every state stabilised by an “AvN triple” of n-qubit Pauli operators yields such a model. The “paradox” half proves much less: an informal identification of Liar cycles with strongly contextual support tables, one exact case (the length-4 Liar cycle is the PR box), and the cohomology of paradox deferred to future work. | Active — read in full 2026-09-27 (NOTE-251); worth reading as the place where the cohomological witness is shown complete for a named, large class (every All-vs-Nothing model, over any ring). It is also where the obstruction is defined as a connecting homomorphism. It does not supersede LIT-277. It re-derives none of LIT-277’s computations, and it drops its negative results: the Hardy false positive and the strongly contextual cover with γ = 0. Its “paradox” title-word is supported by about one page of analogy and one worked identification. |
| LIT-279 v2 | On the Cohomology of Contextuality | Carù (2017), ARXIV-1701.00656. Carù gives a possibilistic model on the two-party, two-setting, four-outcome Bell cover that is strongly contextual, yet whose Čech obstruction vanishes on all 22 support sections (Fig 4, Table 1). That refutes Abramsky–Mansfield–Barbosa’s Conjecture 8.1 for any symmetry or connectedness condition the Bell cover meets. I confirmed the refutation over ℤ and ℤ/2. The rest is weaker. Odd-degree “higher obstructions” form a hierarchy (Prop 5.5), but every level above the first is empty for every no-signalling model (Prop 5.6). Prop 6.3 restates the known torsor description of Ȟ¹. Prop 6.1 is false in the “only if” direction; the PR box refutes it. | Active — read in full 2026-09-27 (NOTE-252); worth reading for one result, the first complete false negative of the cohomological witness (§4 and Appendix A, about two pages), which is correct as I verified it. §5 proves its own programme empty for no-signalling models, and §6 is a standard correspondence plus one false proposition. The follow-up, arXiv 1807.04203, may supersede it; that is unverified, since it was not read. |
| LIT-280 v2 | Towards a complete cohomology invariant for non-locality and contextuality | Carù (2018), ARXIV-1807.04203. For cyclic scenarios, where the contexts’ intersection graph is a single chordless N-cycle, Carù proves that ℤ/2 Čech cohomology of the (N−1)-th iterated “joint model” is a complete invariant for logical and strong contextuality (Thm 7.7). I confirmed this on the Hardy model, on his own 2017 counterexample (detected on all 22 sections at level 3) and on 659 random contextual cycle models. The extension beyond cycles does not hold as stated. Prop 8.2 is false on the paper’s own Table 5 model. Worse, the one non-cyclic case the paper claims to repair, the §8 Kochen–Specker cover of Abramsky–Mansfield–Barbosa, is never detected at any level under the paper’s definitions (my proof and computation). That refutes the paper’s closing Conjecture 9.1. | Active — read in full 2026-09-27 (NOTE-253); worth reading for §§4–7, a correct and genuinely new result: a complete cohomological test for contextuality on cyclic scenarios, which closes the Hardy and Carù-2017 gaps. Read it knowing that §8’s generalisation, its §8.1 Kochen–Specker example and Conjecture 9.1 are wrong as stated. |
| LIT-281 v2 | Logical and Topological Contextuality in Quantum Mechanics and Beyond | Carù (2019), DOI-10.5287/ora-kznbp4k4y. The thesis’s cohomology chapters republish the 2017 and 2018 papers with the “joint model” renamed the “line model”. The definitions, Thm IV.32 (complete on cyclic scenarios at level |M|−1), Prop IV.34, Thm IV.36 and Conjecture IV.41 carry over unchanged. So does every defect found in the preprint. Under the thesis’s own Def IV.26, the §8 Kochen–Specker cover of Abramsky–Mansfield–Barbosa is still detected on only 6 of 15 sections at levels 0–3 (my recomputation), and a lifted ℤ/2 family shows it is never fully detected at any level. The context-wise reading the worked examples use detects all 15 sections at level 2. The genuinely new material lies outside the cohomology programme: a probabilistic line-model construction (§IV.7), a valuation-algebra “theory of disagreement” with inference algorithms (Ch V, bound O(k^n·l^{k(n−1)+1}) on (n,k,l) scenarios), and the AvN-triple characterisation of stabiliser states (Ch VI). | Active — read in full 2026-09-27 (NOTE-254); worth reading as the consolidated source for Carù’s cohomology programme (Chs III–IV), for the valuation-algebra reformulation of contextuality (Ch V) and for the AvN triple theorem (Ch VI). Chs III–IV repeat, uncorrected, the errors recorded for LIT-279 and LIT-280, so read them with NOTE-252 and NOTE-253 at hand. |
| LIT-298 v2 | The problem of hidden variables in quantum mechanics | Kochen & Specker (1967), DOI-10.1512/iumj.1968.17.17004. Theorem 1 exhibits a finite partial Boolean algebra D, generated by 117 directions in E³, that admits no homomorphism onto Z₂. §4 embeds D in the projections of the spin-1 space of orthohelium’s 2³S₁ state, via α ↦ P J_α². So no assignment of 0/1 values to those quantum propositions respects all the Boolean relations within each commeasurable set. Theorem 0 makes the target precise: a partial Boolean algebra embeds in a Boolean algebra iff its Z₂-homomorphisms separate points. That is, hidden variables in the paper’s sense need enough “global valuations”, and quantum mechanics in dimension 3 has none. The same paper builds a classical hidden-variable model for spin ½ (§6). It argues that von Neumann’s no-go needed additivity on non-commuting observables (Theorem 3). | Active — read in full 2026-09-29 (NOTE-279); worth reading. It is the primary source of the Kochen–Specker theorem and of the partial-algebra formulation that the sheaf and topos programmes generalise. The 117-vector construction is superseded for economy by later small KS sets (e.g. Peres’s 33, Cabello’s 18 in dimension 4); the review LIT-263 covers those. |
| LIT-299 v2 | A topos foundation for theories of physics: III. The representation of physical quantities with arrows δ̆°(A): Σ̲ → ℝ̲≽ | Döring & Isham (2008), DOI-10.1063/1.2883777. Each bounded self-adjoint  becomes an arrow δ̆^o(A) : Σ → ℝ^⪰, where ℝ^⪰ is the presheaf of real-valued order-reversing functions on ↓V. At λ ∈ Σ_V it sends V′ ⊆ V to λ(δ^o(Â)_{V′}), and δ^o(Â)_V is the spectral-order-least operator in V above Â. The map  ↦ δ̆^o(A) is injective, but ℝ^⪰ is only an additive monoid, not the real-number object. Its Grothendieck completion k(ℝ^⪰) is an abelian-group object whose global elements are exactly the functions of bounded variation on V(H). | Active — read in full 2026-09-29 (NOTE-290); worth reading as the source of the quantity-value object ℝ^⪰ and of the explicit statement that Gel’fand points of a context are ultrafilters of its projection lattice. Its results are small, mostly proved or cited from de Groote and Döring. The physically motivated parts (k(ℝ^⪰), “intrinsic dispersion”, unitaries) are definitions with no consequences drawn. |
| LIT-306 v2 | The logic of quantum mechanics | Birkhoff & Neumann (1936), DOI-10.2307/1968621. The paper argues heuristically that the experimental propositions of a quantum system correspond to closed linear subspaces of Hilbert space. Meet is intersection, join is closed linear span, and negation is orthogonal complement. The lattice satisfies L1–L4 and complementation L71–L73 but not the distributive law L6. The paper proposes the modular law L5 as its replacement. Under a finite-dimension assumption, it concludes that the propositional calculus of QM is an abstract projective geometry. The Appendix proves that orthocomplementations on P_{n−1}(F), n ≥ 4, are exactly the polarities of definite diagonal Hermitian forms with respect to an involutory anti-automorphism of F. | Active — read in full 2026-09-29 (NOTE-275); worth reading as the founding text of quantum logic, and for its proposed axiom, which is modularity, not orthomodularity. For the orthomodular Hilbert-lattice machinery that is used today (infinite dimensions, Piron, Solèr), later work supersedes it. The paper itself calls its main conclusions “admittedly heuristic”. |
| LIT-310 v2 | A topos foundation for theories of physics: IV. Categories of systems | Döring & Isham (2008), DOI-10.1063/1.2883826. Proposes axioms for a category Sys of physical systems, symmetric-monoidal under composite ⋄ and under disjoint sum ⊔, whose arrows are translations of local languages. Each arrow j : S₁ → S is realised by a left-exact functor τ(j) between the systems’ topoi, together with arrows φ(j) : Σ_{S₁} → τ(j)Σ_S and β(j) : τ(j)R_S → R_{S₁}. Classical physics satisfies the scheme exactly. In quantum theory the disjoint sum H₁⊕H₂ satisfies it (the pulled-back arrow of Â₁⊕Â₂ is δ̆(A₁)), but the composite H₁⊗H₂ does not: the natural pull-back of δ̆(A₁) differs from δ̆(A₁⊗1̂) off the image of V(H₁), which the authors attribute to “operator entanglement”. | Proposed — read in full 2026-09-29 (NOTE-285); promising and unproven. The rules are explicitly “experimental”, the quantum composite fails the commutativity condition the rules state, and the key negative claims ((4.46), and that no operator B̂ gives the pulled-back arrow) are asserted, not shown. What would settle it is a proof of (4.46) and of the non-existence of B̂, plus a variant of the rules (or of the context category) under which tensor-product composites satisfy (3.18). |
| LIT-312 v2 | A theory of concepts and their combinations II: A Hilbert space representation | Aerts & Gabora (2005), DOI-10.1108/03684920510575807. Aerts & Gabora embed the “pet” data in ℂ¹⁴⁰⁰. Each of 1,400 “basic contexts” is one vector of an orthonormal basis, the ground state is the uniform superposition, and each context is the projector onto the span of its basic contexts. The Born rule then returns the measured frequencies as counting ratios, for example cat under “chewing a bone” = 75/303 = 0.25. “Pet fish” is modelled as the entangled state Σ_{u∈E} |u⟩⊗|u⟩/√100 in ℂ¹⁴⁰⁰⊗ℂ⁴⁰⁸, so that collapsing “pet” to goldfish collapses “fish” to goldfish too. Every projector in the model is diagonal in one basis, so the fitted model is a classical probability space in quantum notation, which the authors half-concede (a “quantization … of this set theoretical model”, §3.3). Two of the chosen counts are infeasible: parrot and guppy. | Active — read in full 2026-09-29 (NOTE-299); worth reading as the concrete source of the quantum-cognition treatment of concepts (projectors for contexts, collapse for context change, tensor product and entanglement for combination), and as an early proposal of tensor-product sentence meaning before DisCoCat. Read with the caveat that its worked model never uses a non-commuting projector, so none of its fits exercises anything beyond Kolmogorov probability, and its “solution” to the pet-fish problem reproduces the single-concept data it was given rather than predicting a combination. |
| LIT-313 v2 | On the imbedding of normed rings into the ring of operators in Hilbert space | Gelfand & Neumark (1943), DOI-10.1090/conm/167/16. Theorem 1 is the first abstract characterisation of C*-algebras. Every normed -ring (a unital Banach -algebra with ‖xx‖ = ‖x‖·‖x‖, ‖x*‖ = ‖x‖, and e + x*x invertible) is isometrically *-isomorphic to a norm-closed -subalgebra of the bounded operators on some, not necessarily separable, Hilbert space. The proof builds, for each maximal left ideal M, a positive functional f vanishing on M. The inner product is (ξ, η) = f(yx) on R/M, and the construction takes the direct sum over all M. This is the GNS construction. Lemma 1 is the commutative case: such an algebra is isometrically *-isomorphic to C(𝔐) for its compact maximal-ideal space. §4 adds that the simple weakly closed rings on separable Hilbert space are exactly the factors of types I_n, II₁ and III (Theorem 3). | Active — read in full 2026-09-29 (NOTE-303); worth reading as the primary source of both Gelfand–Naimark theorems, commutative (Lemma 1) and non-commutative (Theorem 1), and of the GNS construction in embryo. It is compact and readable. Modern treatments state it with the single C*-identity. The authors themselves note that it suffices, and later work showed their extra axioms 5′ and 6′ redundant (not verified here). |
| LIT-316 v4 | Quantum Models of Cognition and Decision | Busemeyer & Bruza (2012), DOI-10.1017/CBO9780511997716. Per publisher description: a book-length introduction to modelling human cognition and decision-making with the probability and dynamics of quantum theory, applied to judgment paradoxes, decision-making, concept combination and memory, as an alternative to classical (Kolmogorov) probability models. | Unavailable — filed 2026-09-29 from metadata and published descriptions, not read: no lawful full text has been reachable. Tried again on 2026-10-09 at the owner’s request (manuscript bibliography of that day): OpenAlex still reports it closed, the Internet Archive and HathiTrust hold no copy, Open Library lists no e-book, and Cambridge’s excerpt PDF reset the connection. No NOTE is filed. It stays Unavailable until a copy is supplied. |
| LIT-321 v3 | Local Quantum Physics: Fields, Particles, Algebras (2nd ed.) | Haag (1996), DOI-10.1007/978-3-642-61458-3. Per publisher description: a monograph on algebraic (local) quantum field theory, treating physics through nets of local observable algebras, including the theory of superselection sectors and charges. | Unavailable — seeded 2026-09-29 from metadata and published descriptions, not read: registered because the owner’s prior-art novelty map cites it. A close reading was attempted on 2026-09-29 and no legitimate full text could be reached; it stays Unavailable until a copy is supplied. |
| LIT-325 v3 | A topos perspective on the Kochen–Specker theorem: I. Quantum states as generalized valuations | Isham & Butterfield (1998), DOI-10.1023/A:1026680806775. Isham & Butterfield show that, for bounded self-adjoint operators with discrete spectrum, a global section of the spectral presheaf Σ(Â) = σ(Â) on the category O_d, whose arrows are functional relations B̂ = f(Â), is literally a valuation obeying the value rule and FUNC. A global section of the dual presheaf D(W) = Hom(W, {0,1}) on the poset W of Boolean subalgebras of projectors is literally a Kochen–Specker 0/1 assignment. The Kochen–Specker theorem (cited, not reproved) is therefore equivalent to “no global sections” when dim H > 2 (§§2.2–2.3). In place of the missing global valuations, they construct contextual truth values in the Heyting algebra of sieves: ν^ρ(A ∈ Δ) = {f: B̂ →  | tr(ρ Ê[B ∈ f(Δ)]) = 1}. For every density matrix these satisfy functional composition, null, monotonicity, exclusivity and unit conditions (§4.4, §5.3, Theorem 5.1). | Active — read in full 2026-09-29 (NOTE-271); worth reading as the founding paper of the topos approach to quantum theory and the origin of “Kochen–Specker = no global section of a presheaf over contexts”, credited as such by Abramsky & Brandenburger (LIT-016). The reformulation itself is definitional and its proof is two paragraphs. The paper’s own contribution is the sieve-valued generalized valuation, whose later development (von Neumann algebras as base, daseinisation) lives in Parts III–IV and in Döring–Isham (LIT-343, LIT-335, LIT-299, LIT-310). |
| LIT-335 v2 | A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory | Döring & Isham (2008), DOI-10.1063/1.2883742. A projector P̂ is mapped to the clopen sub-object δ(P̂) of the spectral presheaf Σ over the poset V(H) of abelian von Neumann subalgebras, by “daseinisation” (de Groote’s V-support): δ(P̂)V = the least projector in V above P̂. The map is injective and preserves ∨, but only δ(P̂∧Q̂) ⪯ δ(P̂)∧δ(Q̂). This represents the propositional language PL(S) in the Heyting algebra Sub_cl(Σ). A pure state gives the truth object T^ψ, and “A ε Δ” gets the sieve-valued truth value {V′ ⊆ V | ⟨ψ|δ(Ê[A∈Δ]){V′}|ψ⟩ = 1}. | Active — read in full 2026-09-29 (NOTE-292); worth reading as the core reference for “quantum propositions as clopen sub-objects of the spectral presheaf” (daseinisation, Sub_cl(Σ), truth objects). It is a cleaned-up restatement of Isham–Butterfield (LIT-325) plus de Groote’s operation, not a new no-go result. Several proofs are sketched or omitted, and the “no global elements” statement it relies on needs dim H > 2, which it never says. |
| LIT-339 v2 | The intrinsic parity of elementary particles | Wick et al. (1952), DOI-10.1103/PhysRev.88.101. The paper defines a superselection rule: a decomposition of Hilbert space into orthogonal subspaces A, B, C, … between which there are no spontaneous transitions and no measurable quantities with non-zero matrix elements. Relative phases across subspaces, and hence relative parities, are then unobservable. It proves such a rule between integer and half-integer angular momentum, from time reversal with U_AKU_AK = +1 and U_BKU_BK = −1. It postulates, without conclusive evidence, a charge superselection rule. The consequence is that “all Hermitean operators represent measurable quantities” must be abandoned. | Active — read in full 2026-09-29 (NOTE-304); worth reading as the origin of the superselection concept and its first argued instance (univalence). It is short, and its definition is still the one used. The algebraic form (sectors as the centre of the observable algebra) is later work; Haag’s book, LIT-321, is the place for that. |
| LIT-340 v2 | A theory of concepts and their combinations I: The structure of the sets of contexts and properties | Aerts & Gabora (2005), DOI-10.1108/03684920510575799. Aerts & Gabora model a concept as an entity with states that change under contexts, a “state context property system” (Σ, M, L, µ, ν). They report an 81-subject e-mail study in which 7 contexts shift the rated frequency of 14 exemplars of “pet” (for example dog 0.50 under “The pet is chewing a bone” against 0.12 in the unit context) and the applicability of 14 properties. They then postulate that contexts and properties each form a complete orthocomplemented lattice, and call the structure quantum-like because a state can be an eigenstate of e ∨ f without being one of e or of f (eq. 19). No theorem is proved, the lattice is not shown to be non-distributive or orthomodular, and the paper contains no Hilbert-space model and no treatment of combination; both are deferred to Part II. | Active — read in full 2026-09-29 (NOTE-294); worth reading as the founding statement of the state-context-property approach to concepts that the prior-art map cites, and for its pet data. Read it as a programme and a data set, not as a result. The lattice structure is assumed (“we suppose”, §3.2), the orthocomplement is illustrated rather than derived, and the “quantum-like” criterion it gives does not by itself make the lattice non-Boolean. |
| LIT-342 v3 | What is Structural Realism? | Ladyman (1998), DOI-10.1016/S0039-3681(98)80129-5. Distinguishes epistemic from metaphysical (ontic) structural realism and argues that structural realism should be a revision of ontology, not only of what we can know, drawing on quantum physics (per standard description; abstract not retrieved). | Unavailable — seeded 2026-09-29 from metadata and published descriptions, not read: registered because the owner’s prior-art novelty map cites it. A close reading was attempted on 2026-09-29 and no legitimate full text could be reached; it stays Unavailable until a copy is supplied. |
| LIT-343 v2 | A topos foundation for theories of physics: I. Formal languages for physics | Döring & Isham (2008), DOI-10.1063/1.2883740. The programme paper. Its contention is that a theory of a physical system S is a representation of a typed “local language” L(S) (ground types Σ and R, function symbols A : Σ → R) in a topos. Physical quantities become arrows Σ_φ → R_φ, and propositions become sub-objects of Σ_φ, which form a Heyting algebra. Classical physics is the representation in Sets. The paper proves nothing. Its one argument is that the axiom ¬(A ε Δ) ⇔ A ε ℝ∖Δ is not validated by non-Boolean Heyting representations (§3.2.1). | Active — read in full 2026-09-29 (NOTE-277); worth reading as the statement and vocabulary of the Döring–Isham programme (neo-realism, PL(S) and L(S), state object and quantity-value object, representation in a topos). It is discursive and contains no results, and every piece of quantum mathematics is deferred to papers II–III (LIT-335, LIT-299). |
| LIT-357 v2 | Studies on the Foundation of Quantum Mechanics. I | Husimi (1937), DOI-10.11429/ppmsj1919.19.0_766. Husimi derives Birkhoff and von Neumann’s lattice axioms from statistical premises. Implication is P̄ ≤ Q̄ in every state (and requires compatibility), negation is Q̄ = 1 − P̄, and compatible propositions obey classical logic. Meet is obtained from “Ehrenfest’s principle” (additivity of mean values) via the top eigenvalue of aP + bQ (pp. 778–779), and the modular law, equivalently the parallelogram law |P| + |Q| − |P∩Q| = |P∪Q|, is proved under a finite-chain assumption. The orthomodular identity is used but never named or stated as an axiom. It appears as the step “since we admit traditional logic to any classical part … S + P∩Q = R” for R ⊃ P∩Q (p. 784), and as the case “(Q − P) + P = Q provided Q ⊃ P” of a classical identity (p. 780). | Active — read in full 2026-09-29 (NOTE-312); worth reading as the first derivation of the Birkhoff–von Neumann propositional calculus from operational premises (states, mean values, compatibility), and for its physical account of meet and join. It is not, however, a clean citation for “the orthomodular law”. The law is an unnamed lemma resting on “classical logic within compatible sets”, and the paper’s axiomatic target is modularity under a finite-chain assumption. A text that states orthomodularity as an axiom (e.g. Kalmbach, Orthomodular Lattices, 1983; unverified here) would settle row 1’s citation. |
| LIT-361 v2 | Superselection Rules for Philosophers | Earman (2008), DOI-10.1007/s10670-008-9124-z. A supplemental reading for the owner’s prior-art novelty map, where an earlier reading found the map’s citation wrong or unreachable. | Unavailable — seeded 2026-09-29 as a supplemental reading for the owner’s prior-art novelty map; a close reading was attempted and no legitimate full text could be reached |
| LIT-378 | ’What is a Thing?’: Topos Theory in the Foundations of Physics | Döring & Isham (2008), arXiv:0803.0417. A 212-page restatement of the topos programme of papers I–IV. A theory of a system S is a representation of a typed language L(S) in a topos. In quantum theory the topos is presheaves on the poset V(H) of abelian von Neumann subalgebras, propositions are clopen sub-objects of the spectral presheaf Σ via daseinisation, a pure state gives sieve-valued truth values ν(A ε Δ; ψ)V = {V′ ⊆ V | ⟨ψ|δ(Ê[A∈Δ]){V′}|ψ⟩ = 1}, and each bounded self-adjoint  becomes an arrow Σ → ℝ^↔ whose values are monotone intervals over subcontexts. Classical physics fits its axioms for composite systems exactly; quantum theory fits the disjoint sum but not the tensor-product composite. | Active — read in full 2026-10-01 (NOTE-326); worth reading as the one-volume statement of the Döring–Isham topos programme. It is, by the authors’ own account (p. 14), “partly an amalgam” of papers I–IV (LIT-343, LIT-335, LIT-299, LIT-310), with corrections and some new material. The new material is the value of a quantity in a pseudo-state and the inverse-image link between L(S) and PL(S) (§§8.5–8.8, Thm 8.6), the algebra of the candidate quantity-value objects (§9.3), the Boolean-subalgebra base category (§5.5.3), Kochen–Specker as a lifting obstruction (§6.6), and a long speculative §14–15. It adds no new no-go result. Its proofs are as sketchy as the series’ at the same places: Sub_cl(Σ) is Heyting (Thm 16.1) with ⇒ left unwritten, and ι : O → P_clΣ natural and monic is left to the reader. A reader who has read I–IV gains only §§8.5–8.8, 9.3 and 14; a reader new to the programme should start here. |
| LIT-380 v2 | A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category | Hamilton et al. (1999), arXiv:quant-ph/9911020. Hamilton, Isham & Butterfield move the series’ base category from operators (O) and Boolean subalgebras (W) to the poset V of commutative von Neumann subalgebras of B(H). There they restate Kochen–Specker as “the spectral presheaf Σ(V) = Gelfand spectrum of V has no global elements” (dim H > 2, cited, §2.2). They identify the coarse-graining presheaf G(V) = L(V), G(i)(P̂) = inf{Q̂ ∈ L(V₂) | P̂ ≤ Q̂} (Def. 3.1), with the presheaf CloΣ of clopen subsets of the spectra; the isomorphism is asserted, and part IV proves it. A state ρ yields the sieve-valued valuation ν^ρ_V(P̂) = {V′ ⊆ V | ρ(G(P̂)) = 1} (Def. 3.4) and the support projectors inf{P̂ ∈ L(V) | ρ(P̂) = 1}. These form a global element of G and hence an “interval valuation”, a subobject of Σ (§§4.2–4.4). The probability-r valuations give subobjects of G but not global elements. | Superseded — by LIT-382; read in full 2026-10-01 (NOTE-327); covered better by part IV (Butterfield & Isham 2002, read alongside), which restates this paper’s V-framework definitions nearly verbatim, proves the G ≅ CloΣ isomorphism that this paper only asserts and develops its interval valuations into theorems, and by Döring–Isham II (LIT-335), whose spectral presheaf over V(H) and clopen propositions are the mature form; it remains the citation for introducing V as the base category (as LIT-335 credits it). |
| LIT-381 v2 | A Topos Perspective on the Kochen-Specker Theorem: II. Conceptual Aspects, and Classical Analogues | Butterfield & Isham (1998), arXiv:quant-ph/9808067. Butterfield & Isham argue that the sieve-valued “generalised valuations” of part I are not a quantum peculiarity but the natural valuations for any presheaf of propositions on any small category. Three supports are offered. (1) Classical physics: the value presheaf Υ(Ā) = Ā(S) on the category M of measurable functions has global sections (one per microstate, eq. 3.7), yet a macrostate R ⊆ S yields a sieve-valued valuation ν^R(A ∈ Δ) = {f_M | B̄(R) ⊆ f(Δ)} (eq. 3.10) with the same FUNC property as the quantum ν^ρ. (2) A proof that a sieve-valued valuation on any presheaf G obeys generalised FUNC, ν(B, G(f)(d)) = f*(ν(A, d)), iff it is a natural transformation G → Ω, i.e. a subobject of G (Theorem 4.1). (3) An informal argument, explicitly “not a genuine deduction” (§5.2), that if partial truth is fixed by which weakenings of a proposition are totally true, the truth value is a sieve. | Active — read in full 2026-10-01 (NOTE-329); worth reading as the series’ conceptual companion and the only place it states the classical analogue and the partial-truth motivation; the mathematics is light (one short theorem), and the sieve semantics it motivates is carried on in part III, part IV and Döring–Isham (LIT-343, LIT-335). |
| LIT-382 v2 | A Topos Perspective on the Kochen-Specker Theorem: IV. Interval Valuations | Butterfield & Isham (2001), arXiv:quant-ph/0107123. Butterfield & Isham show, over the poset V of commutative von Neumann subalgebras, that sieve-valued valuations and “interval valuations” (assignments of support projectors, or of subsets of Gelfand spectra) determine each other under stated conditions. If supports s(α, V) = inf{P̂ | α_V(P̂) = true_V} form a global element of the coarse-graining presheaf G, then α_{V₁}(P̂) = {V₂ ⊆ V₁ | G(s(α, V₁)) ≤ G(P̂)} is a sieve-valued valuation obeying FUNC (Theorem 3.1). There is an analogue for “tight” subobjects of Σ (Theorem 3.2). Both are satisfied by every state’s valuation ν^ρ. The ordering ≤ (or ⊆) is shown to be the natural relation securing sievehood, null, monotonicity, exclusivity and unit (§4). The review §2.4 also supplies the proof, via state extension, that G ≅ CloΣ: restricting the clopen set of P̂ to V₂ gives the clopen set of its coarse-graining. Part III had only asserted this. | Active — read in full 2026-10-01 (NOTE-328); worth reading as the finished statement of the series’ von Neumann-algebra framework. It proves the G ≅ CloΣ isomorphism (coarse-graining = direct image of clopen regions) that Döring–Isham’s daseinisation relies on, and it shows that a state’s sieve-valued truth values are fixed by its support, a global element of G. Its two theorems are correct but close to definitional, so its value lies in the framework rather than in depth. |
| LIT-389 v2 | Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity | Isham & Butterfield (1999), arXiv:gr-qc/9910005. Isham & Butterfield argue, without proving anything new, that presheaf toposes over a base of “contexts” suit physics because the subobject classifier Ω is fixed by the base. They say this answers the objection that many-valued logics are arbitrary (§2.2), and that it keeps the logic distributive (§1.1). They then sketch seven applications. Five are speculative one-paragraph proposals (frames, observers, causal sets, TQFT, synthetic-differential-geometry time in continuous histories). One (§3.7) recaps their Kochen–Specker results: KS ⇔ the dual presheaf D(Â) = Hom(W_A, {0,1}) on the operator category O has no global sections (dim H > 2), and every state ψ gives a FUNC-obeying sieve-valued valuation ν^ψ(A ∈ Δ) = {f_O : Â → B̂ | Ê[B ∈ f(Δ)]ψ = ψ}. | Active — read in full 2026-10-01 (NOTE-334); worth reading only as the programme’s short, informal manifesto as of 1999, written before parts III–IV appeared. It joins the Kochen–Specker valuations to the continuum scepticism (real-valued quantities, space and probabilities) that the Döring–Isham programme later develops, and the Literalism/Extra-Values framing appears nowhere else in the record. Its one technical section is a recap of KS I–II, and a reader of LIT-325, LIT-381 and LIT-378 loses little by skipping it. |
| LIT-390 v2 | Topos Theory and Consistent Histories: The Internal Logic of the Set of All Consistent Sets | Isham (1996), arXiv:gr-qc/9607069. Isham takes the poset B of all Boolean subalgebras (“windows”) of the history orthoalgebra UP, ordered by coarse-graining, and shows that the presheaf A_d(W) = {α | some d-consistent W′ ⊇ W contains α} is a subobject of the constant presheaf ΔUP, whereas the naive “realisable in W” assignment is not a presheaf at all (§4.1–4.2). The probability claim ⟨α, p⟩ then gets the sieve-valued semantic value {W ⊆ W₀ | ∃ d-consistent W′ ⊇ W with α ∈ W′} if d(α, α) = p and ∅ otherwise (4.16), a global element 1 → Ω in Set^B, so the logic of “all consistent sets at once” is the distributive but non-Boolean Heyting algebra of sieves on B. | Active — read in full 2026-10-01 (NOTE-333); worth reading as the origin of the Isham school’s sieve-valued, coarse-graining truth values, which the Kochen–Specker series names as its motivation (KS I, ref. [13]); it is the only place the construction is applied to consistent histories, it is short and elementary, and its constructions are correct as stated, but read it knowing that the Heyting values measure only whether α can be embedded in a consistent set (the probability d(α, α) is the same in every window), so it says nothing about the conflicts between consistent sets that motivate it. |
| LIT-441 v2 | The Conscious Mind: In Search of a Fundamental Theory | Chalmers (1996), Oxford University Press. Per its introduction, the book argues that consciousness cannot be reductively explained and that materialism is false, and then moves toward a nonreductive theory with fundamental psychophysical laws. Its contents run from supervenience and explanation through the irreducibility argument, naturalistic dualism and the paradox of phenomenal judgment, to coherence principles, absent, fading and dancing qualia, and information. It ends with a defence of strong AI and a no-collapse reading of quantum mechanics. Unread: registered from the author’s posted contents, introduction and jacket copy. | Unavailable — registered 2026-10-02 from the author’s posted table of contents, introduction and jacket copy, not read: no lawful full text could be reached. The author posts only those three, not draft chapters. Its two central arguments for this record are readable in their paper versions, LIT-406 (the hard problem and naturalistic dualism) and LIT-392 (chapter 7’s organizational invariance). It stays Unavailable until a copy is supplied. |
| LIT-627 v2 | At the crossroad of the search for spontaneous radiation and the Orch OR consciousness theory | Derakhshani, Diósi, Laubenstein, Piscicchia & Curceanu (2022), Physics of Life Reviews 42:8–14. By its abstract: a critical analysis of Orch OR against new bounds on the spontaneous radiation that the simplest gravity-related dynamical collapse model predicts, concluding that Orch OR built on that simplest version is “highly implausible in all the cases analyzed”. Unread: registered from the abstract and the reference list. | Unavailable — registered 2026-10-03 from the abstract, keywords and reference list, not read. The article is closed access at Elsevier. Unpaywall, OpenAlex and Europe PMC list no open copy; there is no arXiv version (searched by title, abstract phrase and author pairs); Diósi’s publication list gives the DOI but, unlike his other 2022 papers, no PDF; ScienceDirect returned a bot challenge and CORE a redirect challenge, so I stopped. A human reader with library access, or a copy requested from the authors, would move this to Active. |
| LIT-788 | The Geometry of Causality | Gogioso & Pinzani (2023), arXiv 2303.09017, Part 3 of the trilogy. Empirical models on any cover of a space of input histories are the points of a causaltope, a product of simplices cut by linear causality equations, so causal fractions are linear programs. Causal separability relative to a given ambient space lets entangled or contextually controlled quantum switches witness indefinite order once known no-signalling constraints are imposed, with the causally separable fraction bounded below by the separable local fraction. | Active — skimmed 2026-10-09 (NOTE-624); worth reading as the geometric half of the trilogy and the successor of the withdrawn LIT-813’s polytope and causal-fraction results: the empirical models on any cover of a space of input histories are exactly the points of a “causaltope”, the product of simplices sliced by linear causality equations derived from the space’s topology, so supported fractions and causally separable fractions are linear programs. Causal separability is defined relative to an ambient space rather than to the indiscrete one, which lets quantum switches with entangled or contextual control witness indefinite causal order once known no-signalling constraints are imposed (and not otherwise); the causally separable fraction is bounded below by the separable local fraction, and in the switch examples it tracks, and sometimes equals, the local fraction (“contextual causality”). The BFW model is 0% supported by every causally complete 3-event space and 100% by each of three indefinite orders. Read in part: introduction, causaltopes, standard causal separability and the examples. |
| LIT-795 | Context effects produced by question orders reveal quantum nature of human judgments | Wang, Solloway, Shiffrin & Busemeyer (2014), PNAS 111(26):9431–9436. Tests the QQ equality of LIT-834, that the probability of giving the same answer to two questions does not depend on their order, on 70 national surveys and two laboratory experiments. Across all 66 Pew surveys from 2001–2011 that varied the order of two questions, order effects are significant (p = 0.0004) and the q values are not (p = 0.4625); paired context effects across 72 studies correlate at r = −0.82 along the predicted line of slope −1. | Active — read 2026-10-09 (NOTE-604); worth reading as the large-scale test of the QQ equality: across 66 Pew surveys chosen without regard to their results, question order changes the answers significantly (p = 0.0004 for the distribution of order-effect χ² values), while the probability of answering the two questions alike stays the same (p = 0.4625 for the q values). Read with the caveats that the supporting information, which holds the proof, the per-study table and the χ² tests, was not reachable and was not read, and that the paper’s title claims more than its evidence: the equality is a symmetry constraint that the authors concede a classical model could be built to satisfy, and LIT-264 shows that data satisfying it are noncontextual. |
| LIT-800 | The Combinatorics of Causality | Gogioso & Pinzani (2022–23), arXiv 2206.08911 (v1 “The Topology and Geometry of Causality”). Generalises causal orders to spaces of input histories, sets of partial input assignments on which outputs may depend, so that causal constraints can depend on inputs. Defines causal completeness, tightness and sequential, parallel and conditional composition; proves the maximal complete spaces are the causal switch spaces; and counts 2644 causally complete spaces on 3 binary-input events (19 from definite orders), with about a billion estimated on 4. | Active — read 2026-10-09 (NOTE-635); worth reading as the paper that replaces causal orders by “spaces of input histories”: finite, join-prime sets of partial functions from events to inputs, each history being data on which the output at its tip event may depend. Order-induced spaces are a special case, and input-dependent (dynamical) causal constraints, such as a switch where one party’s input fixes the order of the other two, are first-class. Spaces form a lattice under inclusion of their extended histories; causal completeness (one tip event per history) recovers definite order; causal switch spaces are exactly the maximal complete spaces. On 3 events with binary inputs there are 2644 causally complete spaces in 102 symmetry classes, of which order-induced spaces account for 19; most are non-tight. It is the first of the three papers that supersede LIT-813, and its spaces replace that paper’s defective “locale of inputs”. |
| LIT-808 | The Topology of Causality | Gogioso & Pinzani (2023), arXiv 2303.07148, Part 2 of the trilogy. Extends Abramsky–Brandenburger from no-signalling to arbitrary, possibly dynamical or indefinite, causal structure: causality is continuity in the lowerset topology, causal functions form a presheaf that fails to be a sheaf exactly on spaces with “solipsistic contextuality”, empirical models live on any open cover, and standard empirical models on causal switch spaces (total orders included) are always local, so the Leggett–Garg model is local for its own total order. | Active — read 2026-10-09 (NOTE-612); worth reading as the direct successor of the withdrawn LIT-813’s sheaf construction, now over spaces of input histories (LIT-800) rather than a “locale of inputs”: a joint input–output function is causal when each event’s output depends only on the histories with that event as a tip, causal functions are exactly free assignments of outputs to tips (up to tip-equivalence in non-tight spaces), and an extended function is causal iff it is continuous for the lowerset topology. Causal functions form a separated presheaf that is a sheaf exactly when the space admits no “solipsistic contextuality”, so tight spaces give a sheaf and some dynamical causal structures do not, letting deterministic local data fail to glue. Empirical models are compatible families of distributions over causal functions on any open cover, from the fully solipsistic to the classical; standard empirical models on causal switch spaces, total orders included, are always local. Source of THEORY-171. |
| LIT-813 v2 | The Sheaf-Theoretic Structure of Definite Causality | Gogioso & Pinzani (2021), QPL 2021, EPTCS 343:301–324; withdrawn by the authors in 2024. Extends Abramsky–Brandenburger to a finite causal order. Inputs live on lower sets, sections are causal functions (an output depends only on inputs in its past), and an empirical model is a causal conditional distribution. Locality is a global section, equivalently a mixture of deterministic causal functions (Proposition 21). The models form a polytope cut out by linear causality equations, and pre-orders sketch indefinite causality. The construction’s “locale of inputs” is not a locale, as the authors’ withdrawal says. | Superseded — by LIT-800, LIT-808, LIT-788; read 2026-10-09 (NOTE-629), and superseded by its authors’ trilogy: LIT-800 (The Combinatorics of Causality, arXiv 2206.08911), LIT-808 (The Topology of Causality, 2303.07148) and LIT-788 (The Geometry of Causality, 2303.09017). The authors withdrew it: the arXiv record’s v3 (3 April 2024) states that the paper “has been superseded by arXiv:2206.08911v4, arXiv:2303.07148 and arXiv:2303.09017”, that its Definition 3 (the locale of inputs) “is not fit for purpose”, and that it is “unlikely to be the right reference”. The reading confirms the defect: the meet given in Proposition 5 can leave the poset, and the poset is not distributive, so it is not a locale (NOTE-629 gives a two-event counterexample). The successors replace the locale of inputs with spaces of input histories under the lowerset topology, a genuine topology; the sheaf of causal functions and locality as a global section are redone in LIT-808, and the polytope and the BFW causal fractions are proved in LIT-788. |
| LIT-819 | Getting to the Bottom of Noether’s Theorem | Baez (2020; chapter 2022), ARXIV-2006.14741. In Poisson algebras, complex *-algebras and Banach–Lie algebras, “a generates symmetries of b iff b generates symmetries of a” follows from the antisymmetry of the bracket and uniqueness of solutions; for a bilinear bracket, antisymmetry is the self-conservation principle {a, a} = 0. The nontrivial content is a map from observables to generators: in complex quantum mechanics multiplication by i, and in a unital JB-algebra Alfsen and Shultz’s dynamical correspondence, whose second condition Baez links to “inverse temperature is imaginary time”. | Active — read 2026-10-09 (NOTE-630); worth reading as an account of what Noether’s theorem, in its Hamiltonian form, actually assumes: given unique solutions of the flow equations, “a generates symmetries of b iff b generates symmetries of a” reduces to the antisymmetry of a bilinear bracket, which is equivalent to each observable conserving itself. The content lies in identifying observables with generators; in quantum theory that identification is multiplication by i, and Alfsen and Shultz’s dynamical correspondence recovers it from a JB-algebra under the self-conservation principle and a second condition that Baez reads as “inverse temperature is imaginary time”. The paper proves nothing new, by its own statement. |
| LIT-831 | Contextuality-by-Default 2.0: Systems with Binary Random Variables | Dzhafarov & Kujala (2017), Quantum Interaction 2016, LNCS 10106:16–32. CbD 2.0 for binary random variables: a system is noncontextual when its bunches are compatible with the multimaximal couplings of its connections, couplings in which every subset of a connection is maximally coupled. For binary variables that coupling exists, is unique and has an explicit staircase form (Theorem 1), so subsystems of a noncontextual system are noncontextual (Theorem 3). Cyclic and consistently connected systems are judged as in CbD 1.0, and the total-variation measure carries over. For non-binary variables the construction fails, and dichotomization is proposed. | Active — read 2026-10-09 (NOTE-643); worth reading as the paper that defines “CbD 2.0”: for systems of binary random variables, the copies of a content across contexts are coupled multimaximally (every subset, equivalently every adjacent pair in the order of their means, maximally coupled) rather than maximally as a whole, and such a coupling exists and is unique (Theorem 1). The change buys heredity (a noncontextual system’s subsystems are noncontextual, Theorem 3) and removes the partial/complete distinction, and it leaves cyclic systems, consistently connected systems and the negative-probability measure as they were. It shows by example that for non-binary variables multimaximal couplings may fail to exist, may not be unique, and are not stable under coarse-graining, and ends by proposing to dichotomize every variable, the step the canonical-systems paper (LIT-790) takes. |
| LIT-834 | A Quantum Question Order Model Supported by Empirical Tests of an A Priori and Precise Prediction | Wang & Busemeyer (2013), Topics in Cognitive Science 5(4):689–710. Models answering an attitude question as projecting a belief state onto the answer’s subspace and replacing the state by the normalised projection. Non-commuting projectors are then necessary for order effects, and a parameter-free constraint follows for any state and any projectors: the probability of answering the two questions differently is the same in both orders (the QQ equality). Five of six data sets meet it; the sixth, where background information was read out between the questions, fails it, as predicted. | Active — read 2026-10-09 (NOTE-636); worth reading as the source of the quantum question (QQ) model and its QQ equality: if each answer is a Lüders projection of one belief state, the probability of giving different answers to two questions is the same in both orders, whatever the state, the projectors or the dimension. Six data sets bear it out, and the Rose–Jackson poll fails it, as the authors predicted, because new information was inserted between the questions. Read with the caveat that the case against classical rivals is thin: the Bayesian argument is informal and the Markov proof is “available upon request”, so the paper shows that unconstrained classical models need not satisfy the equality, not that none can. |
| LIT-836 | Probabilistic Foundations of Contextuality | Dzhafarov & Kujala (2017), Fortschr. Phys. 65:1600040. Being jointly distributed is transitive in Kolmogorovian probability. So if a measurement were the same random variable in every context, overlapping contexts would force a joint distribution, and Kochen–Specker and Bell contradictions would show that such variables do not exist. The assumption to drop is “Noncontextual Identification”. Contextuality becomes the impossibility of a coupling of context-indexed variables in which same-content copies satisfy a property C. For inconsistently connected binary systems, C is multimaximality (Definition II.1). | Active — read 2026-10-09 (NOTE-623); worth reading as the short foundational argument for Contextuality-by-Default. The paper argues that the traditional reading of a Kochen–Specker or Bell-type contradiction (no joint distribution exists) cannot be right within Kolmogorovian probability, because being jointly distributed is transitive, so overlapping contexts already force one. The assumption a reductio refutes is “Noncontextual Identification”, that a content is the same random variable in every context. The paper then restates contextuality as the impossibility of a coupling with a specified property C. It gives multimaximality as C for binary measurements, with the pairwise characterization (Theorem II.3) that CbD 2.0 (LIT-831) cites. It ends with Specker’s three boxes treated without assuming consistent connectedness. |
| LIT-842 | On the Quantum-like Contextuality of Ambiguous Phrases | Wang, Sadrzadeh, Abramsky & Cervantes (2021), ARXIV-2107.14589, SemSpace 2021 (ACL Anthology, pp. 42–52). Treats each word of a two-word ambiguous phrase as a measurement whose outcome is the selected meaning. One hand-specified support model is possibilistically contextual; its BNC/ukWaC probabilities, like all the corpus models, are signalling. Under Contextuality-by-Default two subject–verb/verb–object pairs come out contextual, but the paper’s own bootstrap leaves both plausibly noncontextual. | Active — read in full 2026-10-09 (NOTE-645); worth reading as the first paper to test natural-language data for contextuality in the formal sense rather than the loose one, and as the paper that names the obstacle the test meets: corpus estimates of meaning selection are almost always signalling. In the sheaf-theoretic framework one hand-specified support model ({tap, box} × {pitcher, cabinet}) is possibilistically contextual (a Hardy-type failure), but its corpus probabilities are signalling, so the framework cannot judge them. Under Contextuality-by-Default the Bell-type corpus systems are noncontextual, and two noun–verb pairs read as subject–verb and verb–object (adopt/boxer, throw/pitcher) are CbD-contextual, by 1/30 and 7/30. Its own parametric bootstrap then gives those two a probability above .56 and above .08 of being noncontextual, on fractions with denominators between 3 and 30, so the contextual verdicts are not statistically secured. |
| LIT-843 | Combining contextuality and causality: a game semantics approach | Abramsky, Barbosa & Searle (2024), ARXIV-2307.04786, Phil. Trans. R. Soc. A 382:20230002. Adds an enabling relation to a measurement scenario and reads contextuality as a game: the Experimenter chooses measurements, Nature chooses outcomes, and Nature’s deterministic strategies replace the event sheaf. Recovers flat contextuality and Gogioso–Pinzani causal Bell scenarios, and expresses adaptive MBQC by Experimenter strategies. The strategy presheaf is not a sheaf: compatible deterministic strategies can fail to glue or glue non-uniquely. | Active — read in full 2026-10-09 (NOTE-646); worth reading as the version of sheaf-theoretic contextuality in which measurements are performed in a causal order and the choice of the next one may depend on earlier outcomes. A causal measurement scenario adds an enabling relation (Kahn and Plotkin’s information matrices renamed); Nature’s deterministic strategies over it replace the event sheaf, and the Abramsky–Brandenburger definitions of empirical model and contextuality then go through unchanged. Flat scenarios and Gogioso and Pinzani’s Bell scenarios with a causal order come out as special cases, and Experimenter strategies express adaptive measurement, as in the Anders–Browne GHZ OR-gate. Its sharpest point is that the strategy presheaf is not a sheaf: deterministic, compatible local strategies can fail to glue (Example 7.2) or glue non-uniquely (Example 7.3). A short paper; several results are deferred to an extended version. |
| LIT-844 | Quantum-Like Contextuality in Large Language Models | Lo, Sadrzadeh & Mansfield (2024), ARXIV-2412.16806, Proc. R. Soc. A 481:20240399 (2025). Instantiates a three-sentence anaphora schema with the support of a PR prism 51,966,480 times from Simple English Wikipedia and takes referent probabilities from BERT: 0.148% of the models are sheaf-contextual by a signalling-corrected test and 71.1% are CbD-contextual. The support is fixed by the schema, so each verdict turns on how uncertain BERT is in three sentences. | Active — read in full 2026-10-09 (NOTE-648) from the arXiv preprint (the journal version could not be reached); worth reading as the large-scale test of language-model output for contextuality, and for what its numbers rest on. A PR-prism schema of three sentences (“It is X1 and the same one is X2”, ”… X2 … same … X3”, ”… X3 and the other one is X1”) is instantiated with 866,108 Simple English Wikipedia noun pairs and their adjectives; BERT gives one referent probability per sentence, and the support is fixed by the schema. Of 51,966,480 models, 77,118 (0.148%) pass the signalling-corrected sheaf test (SF < 1/6) and 36,938,948 (71.1%) the CbD test (Δ < 2). Both verdicts are fixed by three numbers per instance; the CbD one holds whenever BERT is uncertain and roughly consistent across the three sentences. The link to embedding distance is a softmax identity plus weak correlations (R² ≤ 0.08). |
| LIT-845 | Closing Bell: Boxing black box simulations in the resource theory of contextuality | Barbosa, Karvonen & Mansfield (2021), ARXIV-2104.11241, in Samson Abramsky on Logic and Structure in Computer Science and Beyond (Springer, 2023). Characterises which convex maps between the empirical models of two scenarios are induced by classical (non-adaptive) procedures: exactly those given by a non-contextual model of a “hom” scenario [S, T] whose outcomes are procedures. Non-local games are procedures into a one-bit scenario, and the hom construction makes scenarios with predicates a closed category. | Active — read in full 2026-10-09 (NOTE-650); worth reading as the result that says which transformations between the empirical models of two different scenarios are classical. A probabilistic procedure S → T (a mixture of deterministic procedures, each answering every measurement of T by a context of S and a post-processing of outcomes) induces a convex map EMP(S) → EMP(T). A convex map is so induced exactly when it comes from a non-contextual model, satisfying a simpliciality predicate, on a new “hom” scenario [S, T] whose outcomes are procedures. Compatible local procedures that do not glue into one global procedure are contextual models of [S, T]: “contextual simulations”. The construction makes scenarios with predicates a closed category. Non-adaptive only; the arXiv v2 corrects Theorem 44 of the published chapter. |
| LIT-846 | Categories of Empirical Models | Karvonen (2018), ARXIV-1804.01514, QPL 2018, EPTCS 287:239–252 (2019). Defines simulations between empirical models on different scenarios: a simplicial relation from target measurements to sets of source measurements, and a stochastic outcome map natural in the context. Non-contextual models are those simulable from the terminal model, the non-contextual fraction is monotone along simulations, strong contextuality is reflected, and contextual models cannot be cloned. | Active — read in full 2026-10-09 (NOTE-651); worth reading as the paper that first gives empirical models on different measurement scenarios a notion of morphism: a simulation d → e answers each measurement of e by a set of jointly measurable measurements of d (a simplicial relation, so the cover may change) and a stochastic map of outcomes natural in the context. With it, non-contextual models are exactly those simulable from the terminal (empty) model, the non-contextual fraction can only grow along a simulation, strong contextuality is reflected backwards, Graham reductions are simulations (Vorob’ev’s theorem), and no contextual model can be cloned. It is restricted to no-signalling models and has no preprocessing. |
| LIT-847 | A comonadic view of simulation and quantum resources | Abramsky, Barbosa, Karvonen & Mansfield (2019), ARXIV-1904.10035, LICS 2019. Measurement protocols form a comonad on empirical models; a simulation of e by d is a co-Kleisli map from d ⊗ c (c non-contextual) to e. Simulability coincides with convertibility by a term in the free operations of the resource theory of contextuality, including a new conditional measurement; the non-contextual fraction is monotone, and contextual models cannot be cloned. | Active — read in full 2026-10-09 (NOTE-647); worth reading as the paper that makes simulation between empirical models adaptive and shows it is the same thing as convertibility by free operations. Measurement protocols (decision trees of compatible measurements) form a comonad MP on scenarios and on empirical models; a simulation of e by d is a deterministic map MP(d ⊗ c) → e with c non-contextual. Such a simulation exists exactly when e is obtained from d by a term in the free operations (relabelling and translation of measurements, coarse-graining, mixing, choice, tensor, and the new conditional measurement), with an equational theory and normal forms. Along any simulation the non-contextual fraction cannot fall, and no contextual model can be cloned. |
| LIT-851 | Analysing Ambiguous Nouns and Verbs with Quantum Contextuality Tools | Wang, Sadrzadeh, Abramsky & Cervantes (2021), Journal of Cognitive Science 22(3):391–420. Proves that in a binary cyclic system CbD’s signalling quantity Δ is twice the summed minimal direct influences of context in Jones’s causal canonical models, and measures it in 90 rank-2 noun–verb systems from British corpora: Δ is about 1.35 whatever the words’ ambiguity, and homonymous verbs carry about 70% of it against about 50% for polysemous ones, on small counts and a marginal test. | Active — read in full 2026-10-09 (NOTE-654); worth reading as the paper that measures how much signalling there is in natural-language meaning selection, and that gives signalling a causal reading. Its Proposition 1 shows that in a binary cyclic system the CbD quantity Δ is twice the sum, over contents, of the least direct influence of context that any of Jones’s canonical (causal) models of the data must posit, each the total-variation distance between a content’s two marginals. Applied to 90 rank-2 noun–verb systems from the BNC and ukWaC, Δ averages about 1.35 in every class of ambiguity, and homonymous verbs carry about 70% of it against about 50% for polysemous verbs (14 against 55 systems; my recomputation from its appendix gives a Welch t of 2.0, two-sided p ≈ 0.06). It tests no system for contextuality. Read with the caveat that most systems rest on very few corpus occurrences: 51 of the 90 have Δ exactly 0, 2 or 4. |
| LIT-892 v3 | Quantum mechanics near closed timelike lines | Deutsch (1991), Phys. Rev. D 44, 3197. The standard model of quantum systems near closed timelike curves: the state on the loop must be a fixed point of the interaction, as a density matrix, and such a fixed point always exists. Not read: registered to ground the record’s contrast between causal-loop consistency and contextuality. | Unavailable — filed 2026-10-10 from its Crossref record, not read: it is registered to ground one claim (CLAIM-141) on what it is standardly cited for, the density-matrix consistency condition for closed timelike curves. No NOTE is filed. It stays Unavailable until it is read, not on merit. Looked for on 2026-10-10: the publisher’s page (APS) refused the request, and the only full text found online is a scan of the journal article on a personal site (gwern.net) with no sign of the publisher’s permission, so it was not used. |
| LIT-934 v2 | Quantum structure in cognition | Aerts (2009), Journal of Mathematical Psychology 53(5):314–348. Membership weights for pairs of concepts and their conjunctions and disjunctions (Hampton 1988) have a Kolmogorovian model iff they are neither over- nor underextended and a Kolmogorovian factor is non-negative; many items fail. A subspace lattice with meet and join fits the second kind of failure but provably not the first, which the paper models with an emergent-concept sector in Fock space. Negation appears only as an orthocomplement used to derive conjunction from disjunction. | Active — read 2026-10-14 (NOTE-808) in full; worth reading as the paper that sorts Hampton’s concept-combination data into two kinds of non-classicality and proves that the main kind, over- and underextension, cannot come from a subspace lattice’s meet and join. Read it knowing that its fitting model has a free parameter per item and stipulated conditions, that its two-layer account of thought is interpretation, and that negation enters only as the orthocomplement used to derive the conjunction model, never measured. Registered for QUESTION-055; THEORY-255 is filed from it. |
| LIT-945 v3 | Unscrambling the omelette of causation and inference: The framework of causal-inferential theories | Schmid et al. (2020). The paper introduces causal-inferential theories, which pair a theory of causal influences with a theory of inference and say how the two interact. In this frame it recasts operational and realist theories and characterizes the assumptions behind no-go theorems based on Bell locality and on generalized noncontextuality. It shows that the existence of a Leibnizian classical realist representation implies a generalized-noncontextual one; the converse is conjectured. Read in part. | Active — read 2026-10-14 (NOTE-737) in part, not in full: the sections that characterise generalized noncontextuality and Leibnizianity (§§V, VI, VIII, IX) were read closely, §II–IV and the proofs in Appendices C–G were not, so the NOTE is Skimmed. Worth reading as the current statement of what a generalized-noncontextuality no-go denies: a classical realist representation that preserves inferential equivalence. It also shows why a prepare-measure scenario, which allows no Bell-like no-go, can still show contextuality. Read it knowing that Leibnizianity implies generalized noncontextuality only in one direction (the converse is Conjecture 1′), that only classical inference is built, that the nonclassical realist theory it aims at is specified by desiderata only, and that it does not discuss the Kochen–Specker or sheaf notions. |
| LIT-980 v2 | What is nonclassical about uncertainty relations? | Catani, Leifer, Scala, Schmid and Spekkens (2022). Some operational theories with nontrivial uncertainty relations admit a generalized-noncontextual model, so such relations alone are not evidence of nonclassicality. For any state whose four-state reflection orbit is realizable (equal predictabilities with flipped signs, and one equivalence of equal mixtures), preparation noncontextuality bounds the predictabilities of two binary measurements by |⟨X⟩| + |⟨Z⟩| ≤ 1. In a theory where every state has the orbit, this bounds the uncertainty relation itself, and qubit theory’s circular trade-off violates it, by up to √2. | Active — read 2026-10-14 (NOTE-825) in full; worth reading as the clearest case where the form of an observed trade-off, not its existence, witnesses preparation contextuality, in a prepare-measure scenario with two binary measurements that are never made together, so that no sheaf or Kochen–Specker test applies. For a state with a realizable four-state reflection orbit, noncontextuality forces |⟨X⟩| + |⟨Z⟩| ≤ 1, with no perfectly distinguishable states needed. Read it knowing that the bound covers a whole uncertainty relation only in a theory where every state has the orbit, that the operational equivalence is used exactly and the claimed noise robustness is asserted, not shown, and that the three-measurement bound is verified by computer only. Filed for QUESTION-067; the account is THEORY-315. |
| LIT-1182 v2 | An Introduction to QBism with an Application to the Locality of Quantum Mechanics | Fuchs, Mermin & Schack (2014). QBism: quantum states and all probabilities, 0 and 1 included, are an agent’s personal judgements. A measurement is “any action an agent takes to elicit a set of possible experiences”, and the outcome is the experience it elicits; it does not reveal a pre-existing state of affairs. The agent models everything external with quantum mechanics, other agents’ reports included, and only her own awareness of her own experience is exempt. Quantum correlations are between one agent’s time-like separated experiences, so the theory is local. | Active — read 2026-10-16 in full (NOTE-955), from the arXiv v1 PDF. Worth reading as the shortest authoritative statement of QBism: states and all probabilities, 1 included, are an agent’s judgements, a measurement is an action the agent takes to elicit experience, and the outcome is that experience. Read it knowing that it keeps judgement (a state assignment) and measurement (an action on an external system) apart, so its loop is judgement, action, experience, update; and that the agent’s own awareness is the one thing it does not model, so it has no measurement of the self. |
| LIT-1190 v2 | The physics of symbols: bridging the epistemic cut | Pattee (2001). Evolution requires an epistemic cut between energy-degenerate, rate-independent genetic symbols and the rate-dependent dynamics of construction they control; the same subject–object cut recurs wherever symbols stand for referents, and physics meets it as measurement. Measurement codes a dynamical state into quiescent symbols and control is its converse; both are executed by non-integrable constraints, whose function is lost if they are described by the laws alone (von Neumann’s regress). The cut is epistemic, not ontological, and arises first at the genotype–phenotype control interface. | Active — read 2026-10-16 in full (NOTE-952), from the author-posted text on the Binghamton CASCI site. Worth reading as Pattee’s own retrospective statement of the epistemic cut: measurement codes a dynamical state into quiescent symbols, control is its converse, both are executed by non-integrable constraints, and no dynamical description of those constraints keeps their function. Read it knowing that it is not measurement-first: the primeval cut is at the genotype–phenotype control interface and “control of construction came first”, and measurement presupposes control of the measuring device. The argument rests on von Neumann’s regress and Pattee’s earlier papers; nothing is derived here. |
| LIT-1284 | Consciousness and the Collapse of the Wave Function | Chalmers and McQueen (2021, in Gao ed., Consciousness and Quantum Mechanics, OUP 2022), arXiv:2105.02314. Develops the thesis that consciousness collapses the wave function by taking consciousness to be super-resistant to superposition, measuring it with integrated information theory, and giving it a continuous-collapse dynamics. Simple versions are falsified by the quantum Zeno effect; more complex ones are in principle testable with quantum computers. On a property-dualist reading a consciousness-to-physics law gives consciousness a causal role, and the authors argue that collapse is the causal-closure gap that philosophers’ closure arguments assume is absent. | Active — read in part 2026-10-20 (NOTE-1012); worth reading as the most developed consciousness-collapse proposal and as the source that frames quantum collapse as the gap in causal closure that a property-dualist consciousness could fill: consciousness (phenomenal Q-shape in integrated information theory) is super-resistant to superposition and drives a continuous-collapse dynamics, simple versions are falsified by the quantum Zeno effect, and more complex versions are in principle testable with quantum computers. |
| LIT-1635 v2 | Relational Quantum Mechanics | Rovelli (1996), arXiv quant-ph/9609002. The notion behind the measurement problem is the observer-independent state, as absolute simultaneity was behind the unease with the Lorentz transformations. A measuring system O and an external P who treats S+O unitarily give different and equally correct accounts of one measurement, so a value is a value relative to a system, any physical system can be the observer, and quantum mechanics, taken as complete, describes the information (correlation) systems have about one another. The relation between two views is itself relative to a third system, and there is no description in toto. Systems are not relational: each is characterized by its set of yes/no questions and an information capacity. Two information postulates and a third, superposition postulate are offered as a partial reconstruction of the formalism. | Active — read in full 2026-10-27 (NOTE-1227), in arXiv v2; worth reading as the founding statement of RQM: the two-description analysis of one measurement, values relative to any physical system, no deducible relation between views and no description in toto, and an information reconstruction it calls “not fully successful”. It keeps systems prior to their relational values, as an assumption: each is characterized by its question set W(S) and capacity N, and the decomposition may vary. It has no ontology of events; that is later. |
| LIT-1640 v2 | Quantum Darwinism | Zurek (2009), a review. Decoherence singles out pointer states; quantum Darwinism adds that observers read the environment, and that decoherence leaves many copies of the pointer observable in it. The mutual information between the system and a fraction f of the environment rises steeply to a plateau at the system’s entropy H_S, so R_δ = 1/f_δ disjoint fragments each hold nearly all of it; redundancy is high only for the pointer observable, and Zurek reads it as objectivity: many observers learn the state independently and without disturbing it. A unitarity argument shows only orthogonal states can be repeatably recorded, without the Born rule, which is derived separately from envariance. The measurement problem is called “largely settled”. | Active — read in full 2026-10-27 (NOTE-1229), from the arXiv text; worth reading as the author’s own synthesis of einselection, redundancy and envariance. Its repeatability argument (eq. 5) is clean; its redundancy results are cited model calculations; its step from redundancy to objectivity is an identification, and it does not contain the later caveat about mutual information that the exchange attributed to other work. |
| LIT-1647 v2 | Agential Realism—A Relation Ontology Interpretation of Quantum Physics | Barad (2022). A compact later statement of agential realism as an interpretation of quantum physics, in which relata exist only within and through their constitutive relations. | Unavailable — filed 2026-10-27, not read. A close reading was attempted on 2026-10-27 and no lawful full text was reached: the chapter is closed on Unpaywall (Oxford Handbooks Online), with no open copy listed. It stays Unavailable until a copy is supplied. |
| LIT-1653 v2 | Posthumanist Performativity: Toward an Understanding of How Matter Comes to Matter | Barad (2003). The short statement of agential realism: phenomena, not independent objects, are the primary ontological units; relata do not precede the intra-actions through which they become determinate; apparatuses are open-ended material-discursive practices that enact agential cuts, making some boundaries and properties determinate within a phenomenon and excluding others. | Unavailable — filed 2026-10-27, not read. A close reading was attempted on 2026-10-27 and no lawful full text was reached: Unpaywall reports the publisher’s copy as closed, and OpenAlex lists no open copy and no repository full text. It stays Unavailable until a copy is supplied. Its two most-cited passages (pp. 814 and 815) are read at second hand in NOTE-1230 and NOTE-1220. |
| LIT-1657 v2 | Meeting the Universe Halfway: Quantum Physics and the Entanglement of Matter and Meaning | Barad (2007). The full development of agential realism from Bohr’s philosophy-physics: the inseparability of object and agencies of observation read as ontological, not epistemic; phenomena as the primary units; apparatuses as boundary-making practices that are themselves phenomena; agential separability as exteriority within phenomena; and ethico-onto-epistem-ology, on which practices of knowing are part of the world’s becoming and carry responsibility for the cuts they enact. | Unavailable — filed 2026-10-27, not read. A close reading was attempted on 2026-10-27 and no lawful full text was reached: the book is closed on Unpaywall (Duke University Press), with no open copy listed. It stays Unavailable until a copy is supplied. |
| LIT-1663 v2 | Constructor theory of information | Deutsch & Marletto (2015), Proc. R. Soc. A 471. Laws are stated as which tasks are possible and which impossible. A set of attributes is an information variable when all its permutations and its cloning are possible; distinguishability is defined from that, which gives the recursive definition of distinguishing a base. Conjectured principles (interoperability, consistency of measurement, ensemble distinguishability and others) cover classical information. One further prohibition, disjoint information observables whose union is not one (“superinformation”), yields indistinguishable states, no-cloning, complementarity, deterministic unpredictability, measurement disturbance and locally inaccessible information; quantum information is an instance. | Active — read in full 2026-10-27 (NOTE-1219); worth reading as the cleanest statement of information as a counterfactual property of a substrate, defined from possible permutations and cloning before distinguishability, and as the derivation of quantum information’s distinctive features from one prohibition on top of conjectured principles. It treats instantiating information as an intrinsic, modal property, so it gives no support to relational constitution. |
| LIT-1666 v2 | Stable Facts, Relative Facts | Di Biagio & Rovelli (2021), Found. Phys. 51:30. A fact happens at every interaction and is relative to the system it affects. Facts relative to F are stable for another system W when W may use them in the classical total-probability rule; the rule’s failure is interference. If F is entangled with an environment whose states overlap by at most ε and W does not interact with it, the rule holds for W up to O(ε): stability is approximate and relational, and a fact can be stable for W before it is a fact for W. The ontology is put in relative facts, not stable ones, and the Frauchiger–Renner contradiction and the Local Friendliness violation are read as the classical rule used across contexts where facts are not stable. | Active — read in full 2026-10-27 (NOTE-1226), from arXiv v3, which is the journal’s typeset text; worth reading as the definition of RQM’s stable facts: a fact relative to one system is stable for another when the other may sum over it classically, which decoherence gives to order ε, only for a system that cannot reach the environment. Stability is approximate and relational, not a step toward observer-independence. Short; its one derivation is a partial trace, and inter-observer agreement gets one sentence. |
| LIT-1672 v2 | Barad, Bohr, and quantum mechanics | Faye & Jaksland (2021). Barad’s Bohr is more ontological than Bohr, who took phenomena as experimental arrangements with their outcomes; Rovelli’s relational quantum mechanics does not entail Barad’s relational holism; and agential realism leaves the measurement problem unsolved, since persisting entanglement of apparatus and object does not explain one determinate outcome. Quantum mechanics does not force agential realism. | Active — read in full 2026-10-27 (NOTE-1220); worth reading as the assessment of agential realism from philosophy of physics: Barad’s Bohr is not Bohr’s (phenomena epistemic, Bohr an entity realist, the dualisms presupposed), quantum mechanics does not entail agential realism (Bohmian mechanics; relational quantum mechanics keeps systems prior to interactions), and agential realism has no answer to the measurement problem as it stands. |
| LIT-1676 v2 | Relative Information, Relative Facts | Di Biagio & Rovelli (2025). Relational quantum mechanics restated in Shannon terms. The information about A relative to B is the expected conditional information, I_{A|B} = I_A + I_{A:B}, and A is a fact relative to B when it is maximal. Only commuting variables have a joint distribution, so perspectives belong to commutative subalgebras (“classical subsystems”), not to quantum systems, and a qubit has none. “Every fact is a relative fact”, and in practice a fact is information “high enough” for the resolution needed. As a side result, a von Neumann measurement is continuous growth of the pointer’s information, sin²(ωt)H_A in the model. Merging perspectives, Wigner’s friend, EPR and the extended Wigner’s friend scenario are treated by worked examples. | Active — read in full 2026-10-27 (NOTE-1214), from arXiv v2; worth reading as the quantitative restatement of relational quantum mechanics, with relative facts defined by relative information and held by commutative subalgebras rather than by quantum systems. The definitions are clean. The resolutions it claims (combination problem, Frauchiger–Renner, intersubjectivity) are paragraphs or citations, and every consistency check is computed relative to one outside system. |
| LIT-1869 | Nature’s Queer Performativity | Barad (2012), Kvinder, Køn & Forskning 1–2:25–53, the authorized version of an essay first printed in Qui Parle (2011). It argues that identity, causality, space and time are performed, not given, in nature as well as culture. Lightning, stingray receptor cells, the dinoflagellate Pfiesteria and atoms are offered as “queer critters” whose behaviour a classical ontology of separate, determinate things cannot account for. Her agential realism replaces things with phenomena: an intra-action enacts an agential cut, “a ‘local’ resolution within the phenomenon of the inherent ontological indeterminacy”, which yields agential separability, not absolute separation. The ethical upshot: there are no acts “against nature”, only acts of nature, and accountability is for the cuts that are made. | Active — read in full 2026-10-30 (NOTE-1458), from the journal’s own PDF of the article (CC BY-NC) on tidsskrift.dk (tidsskrift.dk/KKF/article/download/28067/24695/65067), text extracted with pdftotext. This is the version Barad calls authorized; an earlier Qui Parle printing (2011) is, by her note 15, not the version she submitted. Worth reading as the record’s first text of Barad’s own: it states the agential cut in her words, as a local resolution, enacted by an intra-action within a phenomenon, of an inherent ontological indeterminacy, against the Cartesian cut as an inherent distinction. It is an essay, not an argument from data: its physics is the Bohr model and the which-path and quantum-eraser experiments as she reads them, and its other cases come from Kirby and Schrader. |